L  B 


UC-NRLF 


BIENNIAL   REPORT 


OF    THE 


President  of  the  University 


OF    CALIFORN 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

GIKT  OK 


^Accession i).Q/2.5.8 Class 


Teachers'  Manual  to 
Walsh's  Intermediate 
Arithmetic* 


MATHEMATICS  FOR  COMMON  SCHOOLS 

A 

MANUAL  FOR   TEACHERS 


INCLUDING 


DEFINITIONS,  PRINCIPLES,  AND  RULES 

AND   SOLUTIONS  OF  THE   MORE 

DIFFICULT   PROBLEMS 


BY 


JOHN  H.  WALSH 

ASSOCIATE  SUPERINTENDENT  OF  PUBLIC  INSTRUCTION 
BROOKLYN,   N.Y. 


INTERMEDIATE   ARITHMETIC 


BOSTON,  U.S.A. 

D.  C.  HEATH  &  CO.,  PUBLISHERS 
1896 


Lib 


COPYRIGHT,  1895 

BY  JOHN  H.  WALSH 


J.  S.  Gushing  &  Co.  —  Berwick  &  Smith 
Norwood,  Mass.,  U.S.A. 


CONTENTS 

(INTERMEDIATE  ARITHMETIC  MANUAL.) 


I  PAGC 

INTRODUCTORY 1 

Plan  and  scope  of  the  work  —  Grammar  school  algebra  —  Con- 
structive geometry. 

II 

GENERAL  HINTS     .  5 

Division  of  the  work  —  Additions  and  omissions  —  Oral  and 
written  work  —  Use  of  books  —  Conduct  of  the  recitations  —  Drills 
and  sight  work  —  Definitions,  principles,  and  rules  —  Language 
—  Analysis  —  Obj  ecti ve  illustrations  —  Approximate  answers  — 
Indicating  operations  —  Paper  vs.  slates. 

IX 
NOTES  ON  CHAPTER  Six 45 

X 

NOTES  ON  CHAPTER  SEVEN 58 

XI 
NOTES  ON  CHAPTER  EIGHT 68 

XII 
NOTES  ON  CHAPTER  NINE 75 

XIII 
NOTES  ON  CHAPTER  TEN 87 

DEFINITIONS,  PRINCIPLES,  AND  RULES          ....,.! 

ANSWERS ..1 

in 

90258 


MANUAL   FOR  TEACHERS 


INTRODUCTORY 

Plan  and  Scope  of  the  Work,  —  In  addition  to  the  subjects 
generally  included  in  the  ordinary  text-books  in  arithmetic, 
Mathematics  for  Common  Schools  contains  such  simple  work 
in  algebraic  equations  and  constructive  geometry  as  can  be 
studied  to  advantage  by  pupils  of  the  elementary  schools. 

The  arithmetical  portion  is  divided  into  thirteen  chapters, 
each  of  which,  except  the  first,  contains  the  work  of  a  term  of 
five  months.  The  following  extracts  from  the  table  of  contents 
will  show  the  arrangement  of  topics : 

FIEST  AND  SECOND  YEARS 

Chapter  I,  —  Numbers  of  Three  Figures.  Addition  and  Sub- 
traction. 

THIRD  YEAR 

Chapters  II,  and  III.  —  Numbers  of  Five  Figures.  Multipli- 
ers and  Divisors  of  One  Figure.  Addition  and  Subtraction  of 
Halves,  of  Fourths,  of  Thirds.  Multiplication  by  Mixed  Num- 
bers. Pint,  Quart,  and  Gallon;  Ounce  and  Pound.  Roman 
Notation. 

1 


MANUAL    FOR   TEACHERS 


FOURTH  YEAH 


Chapters  IV,  and  V,  —  Numbers  of  Six  Figures.  Multipliers  and 
Divisors  of  Two  or  More  Figures.  Addition  and  Subtraction  of 
Easy  Fractions.  Multiplication  by  Mixed  Numbers.  Simple 
Denominate  Numbers.  Roman  Notation. 


FIFTH  YEAR 

Chapters  VI,  and  VII,  —  Fractions.     Decimals  of  Three  Places. 
Bills.     Denominate  Numbers.     Simple  Measurements. 


SIXTH  YEAR 

Chapters  VIII,  and  IX,  —  Decimals.     Bills.     Denominate  Num- 
bers.    Surfaces  and  Volumes.     Percentage  and  Interest. 


SEVENTH  YEAR 

Chapters  XI,  and  XII.  —  Percentage  and  Interest.  Commercial 
and  Bank  Discount.  Cause  and  Effect.  Partnership.  Bonds 
and  Stocks.  Exchange.  Longitude  and  Time.  Surfaces  and 
Volumes. 

EIGHTH  YEAR 

Chapters  XIII,  and  XIV,  —  Partial  Payments.  Equation  of 
Payments.  Annual  Interest.  Metric  System.  Evolution  and 
Involution.  Surfaces  and  Volumes. 


INTRODUCTORY  3 

While  all  of  the  above  topics  are  generally  included  in  an 
eight  years'  course,  it  may  be  considered  advisable  to  omit  some 
of  them,  and  to  take  up,  instead,  during  the  seventh  and  eighth 
years,  the  constructive  geometry  work  of  Chapter  XVI.  Among 
the  topics  that  may  be  dropped  without  injury  to  the  pupil  are 
Bonds  and  Stocks,  Exchange,  Partial  Payments,  and  Equation 
of  Payments. 

Grammar  School  Algebra.  —  Chapter  X.,  consisting  of  a  dozen 
pages,  is  devoted  to  the  subject  of  easy  equations  of  one  unknown 
quantity,  as  a  preliminary  to  the  employment  of  the  equation  in 
so  much  of  the  subsequent  work  in  arithmetic  as  is  rendered 
more  simple  by  this  mode  of  treatment.  To  teachers  desirous 
of  dispensing  with  rules,  sample  solutions  of  type  examples,  etc., 
the  algebraic  method  of  solving  the  so-called  "  problems  "  in  per- 
centage, interest,  discount,  etc.,  is  strongly  recommended. 

In  Chapter  XV.,  intended  chiefly  for  schools  having  a  nine 
years'  course,  the  algebraic  work  is  extended  to  cover  simple 
equations  containing  two  or  more  unknown  quantities,  and  pure 
and  affected  quadratic  equations  of  one  unknown  quantity. 

No  attempt  has  been  made  in  these  two  chapters  to  treat 
algebra  as  a  science ;  the  aim  has  been  to  make  grammar-school 
pupils  acquainted,  to  some  slight  extent,  with  the  great  instru- 
ment of  mathematical  investigation,  —  the  equation. 

Constructive  Geometry,  —  Progressive  teachers  will  appreciate  the 
importance  of  supplementing  the  concrete  geometrical  instruction 
now  given  in  the  drawing  and  mensuration  work.  Chapter  XVI. 
contains  a  series  of  problems  in  construction  so  arranged  as  to 
enable  pupils  to  obtain  for  themselves  a  working  knowledge  of 
all  the  most  important  facts  of  geometry.  Applications  of  the 
facts  thus  ascertained,  are  made  to  the  mensuration  of  surfaces 
and  volumes,  the  calculation  of  heights  and  distances,  etc.  No 
attempt  is  made  to  anticipate  tig  work  of  the  high-school  by 
teaching  geometry  as  a  science.  ^ 


4  MANUAL   FOR   TEACHERS 

While  the  construction  problems  are  brought  together  into  a 
single  chapter  at  the  end  of  the  book,  it  is  not  intended  that 
instruction  in  geometry  should  be  delayed  until  the  preceding 
work  is  completed.  Chapter  XVI.  should  be  commenced  not  later 
than  the  seventh  year,  and  should  be  continued  throughout  the 
remainder  of  the  grammar-school  course.  For  the  earlier  years, 
suitable  exercises  in  the  mensuration  of  the  surfaces  of  triangles 
and  quadrilaterals,  and  of  the  volumes  of  right  parallelopipedons 
have  been  incorporated  with  the  arithmetic  work. 


II 

GENERAL  HINTS 

Division  of  the  Work.  —  The  five  chapters  constituting  Part  I. 
of  Mathematics  for  Common  Schools  should  be  completed  by  the 
end  of  the  fourth  school  year.  Chapter  I.,  with  the  additional 
oral  work  needed  in  the  case  of  young  pupils,  will  occupy  about 
two  years;  the  remaining  four  chapters  should  not  take  more 
than  half  a  year  each.  When  the  Grube  system  is  used,  and  the 
work  of  the  first  two  years  is  exclusively  oral,  it  will  be  possible, 
by  omitting  much  of  the  easier  portions  of  the  first  two  chapters, 
to  cover,  during  the  third  year,  the  ground  contained  in  Chapters 
I.,  II.,  and  III.  The  remaining  eight  arithmetic  chapters  consti- 
tute half-yearly  divisions  for  the  second  four  years  of  school. 

Additions  and  Omissions.  —  The  teacher  should  freely  supple- 
ment the  work  of  the  text-book  when  she  finds  it  necessary  to  do 
so ;  and  she  should  not  hesitate  to  leave  a  topic  that  her  pupils 
fully  understand,  even  though  they  may  not  have  worked  all  the 
examples  given  in  connection  therewith.  A  very  large  number 
of  exercises  is  necessary  for  such  pupils  as  can  devote  a  half-year 
to  the  study  of  the  matter  furnished  in  each  chapter.  In  the 
case  of  pupils  of  greater  maturity,  it  will  be  possible  to  make 
more  rapid  progress  by  passing  to  the  next  topic  as  soon  as  the 
previous  work  is  fairly  well  understood. 

Oral  and  Written  Work.  — The  heading  "Slate  Problems"  is 
merely  a  general  direction,  and  it  should  be  disregarded  by  the 
teacher  when  the  pupils  are  able  to  do  the  work  "mentally." 
The  use  of  the  pencil  should  be  demanded  only  so  far  as  it  may 

5 


D  MANUAL    FOR    TEACHERS 

be  required.  It  is  a  pedagogical  mistake  to  insist  that  all  of  the 
pupils  of  a  class  should  set  down  a  number  of  figures  that  are 
not  needed  by  the  brighter  ones.  As  an  occasional  exercise,  it 
may  be  advisable  to  have  scholars  give  all  the  work  required  to 
solve  a  problem,  and  to  make  a  written  explanation  of  each  step 
in  the  solution ;  but  it  should  be  the  teacher's  aim  to  have  the 
majority  of  the  examples  done  with  as  great  rapidity  as  is  con- 
sistent with  absolute  correctness.  It  will  be  found  that,  as  a 
rule,  the  quickest  workers  are  the  most  accurate. 

Many  of  the  slate  problems  can  be  treated  by  some  classes  as 
"  sight "  examples,  each  pupil  reading  the  question  for  himself 
from  the  book,  and  writing  the  answer  at  a  given  signal  without 
putting  down  any  of  the  work. 

Use  of  Books,  —  It  is  generally  recommended  that  books  be 
placed  in  pupils'  hands  as  early  as  the  third  school  year.  Since 
many  children  are  unable  at  this  stage  to  read  with  sufficient 
intelligence  to  understand  the  terms  of  a  problem,  this  work 
should  be  done  under  the  teacher's  direction,  the  latter  reading 
the  questions  while  the  pupils  follow  from  their  books.  In  later 
years,  the  problems  should  be  solved  by  the  pupils  from  the 
books  with  practically  no  assistance  whatever  from  the  teacher. 

Conduct  of  the  Kecitation.  —  Many  thoughtful  educators  consider 
it  advisable  to  divide  an  arithmetic  class  into  two  sections,  for 
some  purposes,  even  where  its  members  are  nearly  equal  in 
attainments.  The  members  of  one  division  of  such  a  class  may 
work  examples  from  their  books  while  the  others  write  the 
answers  to  oral  problems  given  by  the  teacher,  etc. 

Where  a  class  is  thus  taught  in  two  divisions,  the  members  of 
each  should  sit  in  alternate  rows,  extending  from  the  front 
of  the  room  to  the  rear.  Seated  in  this  way,  a  pupil  is  doing  a 
different  kind  of  work  from  those  on  the  right  and  the  left,  and 
he  would  not  have  the  temptation  of  a  neighbor's  slate  to  lead 
him  to  compare  answers. 


GENERAL   HINTS  7 

As  an  economy  of  time,  explanations  of  new  subjects  might  be 
given  to  the  whole  class;  but  much  of  the  arithmetic  work 
should  be  done  in  "sections,"  one  of  which  is  under  the  im- 
mediate direction  of  the  teacher,  the  other  being  employed 
in  "seat"  work.  In  the  case  of  pupils  of  the  more  advanced 
classes,  "seat"  work  should  consist  largely  of  "  problems "  solved 
without  assistance.  Especial  pains  have  been  taken  to  so  grade 
the  problems  as  to  have  none  beyond  the  capacity  of  the  average 
pupil  that  is  willing  to  try  to  understand  its  terms.  It  is  not 
necessary  that  all  the  members  of  a  division  should  work  the 
same  problems  at  a  given  time,  nor  the  same  number  of  prob- 
lems, nor  that  a  new  topic  should  be  postponed  until  all  of  the 
previous  problems  have  been  solved. 

Whenever  it  is  possible,  all  of  the  members  of  the  division 
working  under  the  teacher's  immediate  direction  should  take 
part  in  all  the  work  done.  In  mental  arithmetic,  for  instance, 
while  only  a  few  may  be  called  upon  for  explanations,  all  of  the 
pupils  should  write  the  answers  to  each  question.  The  same  is 
true  of  much  of  the  sight  work,  the  approximations,  some  of  the 
special  drills,  etc. 

Drills  and  Sight  Work.  —  To  secure  reasonable  rapidity,  it  is 
necessary  to  have  regular  systematic  drills.  They  should  be 
employed  daily,  if  possible,  in  the  earlier  years,  but  should  never 
last  longer  than  five  or  ten  minutes.  Various  kinds  are  sug- 
gested, such  as  sight  addition  drills,  in  Arts.  3,  11,  24,  26,  etc. ; 
subtraction,  in  Arts.  19,  50,  53,  etc. ;  multiplication,  in  Arts.  71, 
109,  etc. ;  division,  in  Arts.  199,  202,  etc. ;  counting  by  2's,  3's, 
etc.,  in  Art.  61 ;  carrying,  in  Art.  53,  etc.  For  the  young  pupil, 
those  are  the  most  valuable  in  which  the  figures  are  in  his  sight, 
and  in  the  position  they  occupy  in  an  example ;  see  Arts.  3,  34, 
164,  etc. 

Many  teachers  prepare  cards,  each  of  which  contains  one  of 
the  combinations  taught  in  their  respective  grades.  Showing 
one  of  these  cards,  the  teacher  requires  an  immediate  answer 


8  MANUAL   FOE   TEACHEES 

from  a  pupil.  If  his  reply  is  correct,  a  new  card  is  shown  to 
the  next  pupil,  and  so  on.  Other  teachers  write  a  number  of 
combinations  on  the  blackboard,  and  point  to  them  at  random, 
requiring  prompt  answers.  When  drills  remain  on  the  board 
for  any  considerable  time,  some  children  learn  to  know  the 
results  of  a  combination  by  its  location  on  the  board,  so  that 
frequent  changes  in  the  arrangement  of  the  drills  are,  therefore, 
advisable.  The  drills  in  Arts.  Ill,  112,  and  115  furnish  a  great 
deal  of  work  with  the  occasional  change  of  a  single  figure. 

For  the  higher  classes,  each  chapter  contains  appropriate 
drills,  which  are  subsequently  used  in  oral  problems.  It  happens 
only  too  frequently  that  as  children  go  forward  in  school  they 
lose  much  of  the  readiness  in  oral  and  written  work  they 
possessed  in  the  lower  grades,  owing  to  the  neglect  of  their 
teachers  to  continue  to  require  quick,  accurate  review  work  in 
the  operations  previously  taught.  These  special  drills  follow 
the  plan  of  the  combinations  of  the  earlier  chapters,  but  gradu- 
ally grow  more  difficult.  They  should  first  be  used  as  sight 
exercises,  either  from  the  books  or  from  the  blackboard. 

To  secure  valuable  results  from  drill  exercises,  the  utmost 
possible  promptness  in  answers  should  be  insisted  upon. 

Definitions,  Principles,  and  Rules,  —Young  children  should  not 
memorize  rules  or  definitions.  They  should  learn  to  add  by 
adding,  after  being  first  shown  by  the  teacher  how  to  perform 
the  operation.  Those  not  previously  taught  by  the  Grube 
method  should  be  given  no  reason  for  "  carrying."  In  teaching 
such  children  to  write  numbers  of  two  or  three  figures,  there  is 
nothing  gained  by  discussing  the  local  value  of  the  digits.  Dur- 
ing the  earlier  years,  instruction  in  the  art  of  arithmetic  should 
be  given  with  the  least  possible  amount  of  science.  While  prin- 
ciples may  be  incidentally  brought  to  the  view  of  the  children 
at  times,  there  should  be  no  cross-examination  thereon.  It  may 
be  shown,  for  instance,  that  subtraction  is  the  reverse  of  addition, 
and  that  multiplication  is  a  short  method  of  combining  equal 


GENERAL   HI 


numbers,  etc.  ;  but  care  should  be  ts^f^jgj^jfStiiSe  of  pupils 
below  about  the  fifth  school  year  not  to.  dwell  long  on  this  side 
of  the  instruction.  By  that  time,  pupils  should  be  able  to  add, 
subtract,  multiply,  and  divide  whole  numbers  ;  to  add  and  sub- 
tract simple  mixed  numbers,  and  to  use  a  mixed  number  as  a 
multiplier  or  a  multiplicand  ;  to  solve  easy  problems,  with  small 
numbers,  involving  the  foregoing  operations  and  others  contain- 
ing the  more  commonly  used  denominate  units.  Whether  or  not 
they  can  explain  the  principles  underlying  the  operations  is  of 
next  to  no  importance,  if  they  can  do  the  work  with  reasonable 
accuracy  and  rapidity. 

When  decimal  fractions  are  taken  up,  the  principles  of  Arabic 
notation  should  be  developed  ;  and  about  the  same  time,  or  some- 
what later,  the  principles  upon  which  are  founded  the  operations 
in  the  fundamental  processes,  can  be  briefly  discussed. 

Definitions  should  in  all  cases  be  made  by  the  pupils,  their 
mistakes  being  brought  out  by  the  teacher  through  appropriate 
questions,  criticisms,  etc.  Systematic  work  under  this  head 
should  be  deferred  until  at  least  the  seventh  year. 

The  use  of  unnecessary  rules  in  the  higher  grades  is  to  be 
deprecated.  When,  for  instance,  a  pupil  understands  that  per 
cent  means  hundredths,  that  seven  per  cent  means  seven  hun- 
dredths,  it  should  not  be  necessary  to  tell  him  that  7  per  cent  of 
143  is  obtained  by  multiplying  143  by  .07.  It  should  be  a  fair 
assumption  that  his  previous  work  in  the  multiplication  of 
common  and  of  decimal  fractions  has  enabled  him  to  see  that 
7  per  cent  of  143  is  -j-fo  of  143  or  143  X  .07,  without  information 
other  than  the  meaning  of  the  term  "  per  cent." 

When  a  pupil  is  able  to  calculate  that  15  %  of  120  is  18,  he 
should  be  allowed  to  try  to  work  out  for  himself,  without  a  rule, 
the  solution  of  this  problem  :  18  is  what  per  cent  of  120  ?  or  of 
this:  18  is  15%  of  what  number?  These  questions  should 
present  no  more  difficulty  in  the  seventh  year  than  the  following 
examples  in  the  fifth  :  (a)  Find  the  cost  of  ^  ton  of  hay  at  $12 
per  ton.  (b)  When  hay  is  worth  $12  per  ton,  what  part  of  a 


10  MANUAL   FOR   TEACHERS 

ton  can  be  bought  for  $  1.80  ?  (c)  If  -&  ton  of  hay  costs  $1.80, 
what  is  the  value  of  a  ton  ? 

When,  however,  it  becomes  necessary  to  assist  pupils  in  the 
solution  of  problems  of  this  class,  it  is  more  profitable  to  furnish 
them  with  a  general  method  by  the  use  of  the  equation,  than 
with  any  special  plan  suited  only  to  the  type  under  immediate 
discussion. 

In  the  supplement  to  the  Manual  will  be  found  the  usual  defini- 
tions, principles,  and  rules,  for  the  teacher  to  use  in  such  a  way 
as  her  experience  shows  to  be  best  for  her  pupils.  The  rules 
given  are  based  somewhat  on  the  older  methods,  rather  than  on 
those  recommended  by  the  author.  He  would  prefer  to  omit 
entirely  those  relating  to  percentage,  interest,  and  the  like  as 
being  unnecessary,  but  that  they  are  called  for  by  many  success- 
ful teachers,  who  prefer  to  continue  the  use  of  methods  which 
they  have  found  to  produce  satisfactory  results. 

Language.  —  While  the  use  of  correct  language  should  be 
insisted  upon  in  all  lessons,  children  should  not  be  required  in 
arithmetic  to  give  all  answers  in  "  complete  sentences."  Espe- 
cially in  the  drills,  it  is  important  that  the  results  be  expressed 
in  the  fewest  possible  words. 

Analyses.  —  Sparing  use  of  analyses  is  recommended  for  begin- 
ners. If  a  pupil  solves  a  problem  correctly,  the  natural  inference 
should  be  that  his  method  is  correct,  even  if  he  be  unable  to  state 
it  in  words.  When  a  pupil  gives  the  analysis  of  a  problem,  he 
should  be  permitted  to  express  himself  in  his  own  way.  Set 
forms  should  not  be  used  under  any  circumstances. 

Objective  Illustrations.  —  The  chief  reason  for  the  use  of  objects 
in  the  study  of  arithmetic  is  to  enable  pupils  to  work  without 
them.  While  counters,  weights  and  measures,  diagrams,  or  the 
like  are  necessary  at  the  beginning  of  some  topics,  it  is  important 
to  discontinue  their  use  as  soon  as  the  scholar  is  able  to  proceed 
without  their  aid. 


GENERAL   HINTS  11 

Approximate  Answers,  —  An  important  drill  is  furnished  in 
the  "approximations."  (See  Arts.  521,  669,  719,  etc.)  Pupils 
should  be  required  in  much  of  their  written  work  to  estimate 
the  result  before  beginning  to  solve  a  problem  with  the  pencil. 
Besides  preventing  an  absurd  answer,  this  practice  will  also  have 
the  effect  of  causing  a  pupil  to  see  what  processes  are  necessary. 
In  too  many  instances,  work  is  commenced  upon  a  problem  before 
the  conditions  are  grasped  by  the  youthful  scholar ;  which  will 
be  less  likely  to  occur  in  the  case  of  one  who  has  carefully 
"  estimated  "  the  answer.  The  pupil  will  frequently  find,  also, 
that  he  can  obtain  the  correct  result  without  using  his  pencil 
at  all. 

Indicating  Operations.  —  It  is  a  good  practice  to  require  pupils 
to  indicate  by  signs  all  of  the  processes  necessary  to  the  solution 
of  a  problem,  before  performing  any  of  the  operations.  This  fre- 
quently enables  a  scholar  to  shorten  his  work  by  cancellation,  etc. 
In  the  case  of  problems  whose  solution  requires  tedious  processes, 
some  teachers  do  not  require  their  pupils  to  do  more  than  to 
indicate  the  operations.  It  is  to  be  feared  that  much  of  the  lack 
of  facility  in  adding,  multiplying,  etc.,  found  in  the  pupils  of 
the  higher  classes  is  due  to  this  desire  to  make  work  pleasant. 
Instead  of  becoming  more  expert  in  the  fundamental  operations, 
scholars  in  their  eighth  year  frequently  add,  subtract,  multiply, 
and  divide  more  slowly  and  less  accurately  than  in  their  fourth 
year  of  school. 

Paper  vs.  Slates,  —  To  the  use  of  slates  may  be  traced  very  much 
of  the  poor  work  now  done  in  arithmetic.  A  child  that  finds  the 
sum  of  two  or  more  numbers  by  drawing  on  his  slate  the  number 
of  strokes  represented  by  each,  and  then  counting  the  total,  will 
have  to  adopt  some  other  method  if  his  work  is  done  on  material 
that  does  not  permit  the  easy  obliteration  of  the  tell-tale  marks. 
When  the  teacher  has  an  opportunity  to  see  the  number  of 
attempts  made  by  some  of  her  pupils  to  obtain  the  correct  quo- 


12  MANUAL   FOR   TEACHERS 

tient  figures  in  a  long  division  example,  she  may  realize  the 
importance  of  such  drills  as  will  enable  them  to  arrive  more 
readily  at  the  correct  result. 

The  unnecessary  work  now  done  by  many  pupils  will  be  very 
much  lessened  if  they  find  themselves  compelled  to  dispense  with 
the  "  rubbing  out"  they  have  an  opportunity  to  indulge  in  when 
slates  are  employed.  The  additional  expense  caused  by  the 
introduction  of  paper  will  almost  inevitably  lead  to  better  results 
in  arithmetic.  The  arrangement  of  the  work  will  be  looked 
after ;  pupils  will  not  be  required,  nor  will  they  be  permitted,  to 
waste  material  in  writing  out  the  operations  that  can  be  per- 
formed mentally ;  the  least  common  denominator  will  be  deter- 
mined by  inspection  ;  problems  will  be  shortened  by  the  greater 
use  of  cancellation,  etc.,  etc.  Better  writing  of  figures  and  neater 
arrangement  of  problems  will  be  likely  to  accompany  the  use  of 
material  that  will  be  kept  by  the  teacher  for  the  inspection  of 
the  school  authorities.  The  endless  writing  of  tables  and  the 
long,  tedious  examples  now  given  to  keep  troublesome  pupils 
from  bothering  a  teacher  that  wishes  to  write  up  her  records, 
will,  to  some  extent,  be  discontinued  when  slates  are  no  longer 
used. 


IX 
NOTES  ON  CHAPTER  SIX 

The  previous  work  in  mixed  numbers  should  make  the  pupils 
reasonably  familiar  with  the  addition  and  subtraction  of  frac- 
tions having  small  denominators.  In  this  chapter,  the  work  is 
extended  to  cover  the  addition  and  subtraction  of  fractions  whose 
common  denominator  is  determinable  by  inspection.  For  the 
present,  the  teacher  should  be  satisfied  if  her  pupils  acquire  rea- 
sonable facility  in  performing  the  various  operations,  even  if 
they  are  unable  to  formulate,  in  the  language  of  experienced 
mathematicians,  the  reasons  for  the  different  steps.  The  children 
should  be  required  to  use  correctly  and  intelligently  such  techni- 
cal terms  as  are  required  by  the  work  of  the  chapter ;  but  they 
should  not  be  compelled  to  memorize  any  definitions  that  convey 
to  them  no  meaning.  They  should  incidentally  learn  what  is 
meant  by  numerator,  denominator,  common  denominator,  mul- 
tiple, etc.,  by  hearing  the  teacher  employ  these  words  from  time 
to  time,  rather  than  by  commencing  with  what  is  to  them  an 
unintelligible  jumble  of  words. 

451.  While  systematic  work  in  fractions  belongs  properly  to 
the  next  chapter,  the  teacher  should  not  hesitate  to  call  -^,  *£-, 
etc.,  "  improper  fractions,"  and  to  ask  a  pupil  to  state  how  they 
are  changed  to  whole  or  to  mixed  numbers. 

453.  Do  not,  for  the  present,  formulate  the  rule  for  changing 
a  fraction  to  an  equivalent  one  with  higher  terms. 

458.  The  meaning  of  "  lowest  terms "  is  given  in  No.  6. 
Leave  the  rule  for  the  next  chapter.  After  a  pupil  has  rea- 

45 


46  MANUAL    FOE,   TEACHERS 

soned  out  in  his  own  way  that  18  hours  is  f  day,  in  No.  15,  the 
teacher  may  explain  that  18  hours  can  be  written  -Jf  day,  which 
is  reducible  to  the  answer  given  above. 

463.  Have  pupils  see  that  |-  is  larger  than  either  |-  or  -|, 
because  1  sixth  is  larger  than  a  seventh  or  an  eighth ;  and  this 
for  the  reason  that  the  fewer  the  number  of  equal  divisions  made 
in  a  unit,  the  larger  is  each  portion.  Do  not  require  scholars 
to  change  these  fractions  to  equivalent  ones  having  a  common 
denominator. 

467.  For  finding  the  difference  between  two  mixed  numbers 
when  the  fraction  in  the  subtrahend  is  greater  than  that  in  the 
minuend,  the  method  given  in  the  text-book  is  the  one  generally 
employed.  The  teacher  should  always  consider  herself  at  liberty 
to  use  any  other  way  of  performing  this  and  other  operations, 
but  she  should  not  willingly  adopt  any  method  that  is  more 
tedious.  Children  should  not,  for  instance,  be  required  to  change 
mixed  numbers  to  improper  fractions,  and  then  to  reduce  these 
to  a  common  denominator  in  order  to  subtract  one  from  the 
other. 

469.  Pupils  should  now  be  required  to  pay  more  and  more 
attention  to  the  arrangement  of  the  problem  work,  without, 
however,  being  permitted  to  use  unnecessary  figures  or  to  waste 
time.  In  some  good  schools,  the  full  written  analysis  of  a  prob- 
lem is  occasionally  used  as  an  exercise  in  composition. 

When  the  pupils  find  difficulty  in  determining  the  operations 
necessary  to  the  solutions  of  problems,  the  latter  should  be  used 
as  "  sight "  work.  The  alterations  in  the  figures  needed  to  sim- 
plify a  problem  should  now  be  made  by  a  pupil,  instead  of  by  the 
teacher,  as  recommended  in  previous  chapters.  The  scholar  that 
reads  No.  1,  for  instance,  might  change  5f  and  4-J  yards,  to 
5  and  4,  respectively.  No.  2  can  be  solved  as  it  stands.  In 
No.  3,  $150  might  be  substituted  for  $140.40,  and  $2  for  $1.80. 


NOTES   ON   CHAPTER  SIX  47 

Work  of  this  kind  should  gradually  lead  the  pupil  to  form  the 
habit  of  using  some  similar  method  of  ascertaining  for  himself 
how  to  manage  a  problem. 

471.  Written  in  this  form:  3)93|,  No.  11  should  give  the 
children  no  trouble.  If,  however,  they  hesitate  when  the  frac- 
tion is  reached,  the  difficulty  may  be  cleared  up  by  making  a 
concrete  problem :  Divide  $  93J  equally  among  3  persons. 
What  is  the  share  of  each  ? 

Under  no  circumstances  should  these  dividends  be  changed 
to  improper  fractions. 

484.   In  nearly  all  of  the   previous  multiplication  work   in- 
volving mixed  numbers,  the  latter  have  been  used  as         -,«- 
multipliers.     In  No.  35,  the  mixed  number  appears  as 
a  multiplicand.     The  first  six  of  these  examples  and  the     q  \qi~~ 
last  two  should  be  used  as  sight  work,  the  answers  being         ^Q, 
written  directly  from  the  book.    When  the  pupil  reaches         e^ 
one  that  needs  to  be  worked  out  in  full,  say  No.  41,  he       ,« 
should  not  be  permitted  to  use  1SJ  as  a  multiplier,  as  it 
is  important  that  he  should  learn  the  proper  method  of  working 
both  classes  of  examples. 

489.  Many  scholars  will  carelessly  give  20  halves  as  the 
result  obtained  by  dividing  500  halves  by  25  halves.  To  pre- 
vent the  possibility  of  a  mistake  of  this  kind,  some  teachers 
multiply  the  divisor  and  the  dividend  by  the  least  common 
multiple  of  the  denominators  of  the  fractions.  While  this 

18f)1387£  method  produces  exactly  the  same  figures  as  the 
X  4  x  4  one  given  in  the  text-book,  it  is  probably  less  likely 

75)  5550       to  be  followed  by  the  error  mentioned  above. 

497.  Some  teachers  may  prefer  to  write  the  example  as  is 
here  given,  although  using  5  as  the  multiplier.  Other  $.05 
teachers  "  analyze  "  as  follows :  At  \4  per  lb.,  157  X  157 
pounds  of  sugar  would  cost  $1.57;  at  bf  per  lb.,  the  $7.85 


48  MANUAL    FOR   TEACHERS 

$1.57     cost  is  5  times  $1.57.     Business  men  pay  no  attention 
X  5     to  these  fine-spun  distinctions  ;  they  use  as  a  multiplier 
$7.85     the  most  convenient  number,  and  write  the  dollar  sign 
and  the  period  in  the  product  alone. 

500.  In  analyzing  problems  of  this  kind,  it  is  better,  perhaps, 
to  emphasize  the  fact  that  multiplication  is  employed  in  obtain- 
ing the  result.  Thus,  32  base-balls  @  25^  =  32  times  $  J  = 
32  quarters  =  $8 

502.  In  No.  28,  the  price  of  11  yards  can  be  found  by  taking 
11  times  $|,  or  33  quarters,  etc.  No.  24  is  rendered  easier 
by  saying  that  24  bushels  at  1  quarter  per  bushel  would  cost 
6  dollars ;  and  that  at  3  quarters  per  bushel  the  cost  would  be 
3  times  6  dollars,  etc.  Pupils  should  be  encouraged  to  use  the 
method  best  adapted  to  the  particular  example  under  considera- 
tion. 

509.  See  Art.  306. 

510.  In  finding,  for  example,  the  number  of  50-cent  knives 
that  can  be  purchased  for  $  20,  it  may  be  advisable  to  make  the 
division  idea  prominent.     The  analysis  can  take  some  such  form 
as  this :  There  can  be  bought  as  many  knives  as  one  half-dollar 
is  contained  times  in  20  dollars  —  or,  as  there  are  half-dollars 
in  20  dollars. 

Later  problems  involving  division  of  fractions  cause  less 
trouble  if  the  appropriate  operation  is  always  kept  before  the 
pupils,  regardless  of  the  method  employed  to  shorten  the  solu- 
tion of  questions  of  certain  types.  These  short  methods  should, 
however,  be  used. 

To  ascertain  the  number  of  2-dollar  knives  obtainable  for 
$24,  the  scholar  turns  naturally  to  division;  and  he  should 
learn  to  see  that  he  actually  divides  when  he  obtains  48  as  the 
number  of  50-cent  knives  that  can  be  purchased  for  the  same 


NOTES  ON   CHAPTER   SIX  49 

money.     In  the  latter  case,  the  numbers   given  are  24  and  £, 
from  which  48  can  result  only  when  £  is  used  as  a  divisor. 

Many  pupils  that  give  the  correct  answer  when  24  -s-  £  is 
placed  upon  the  blackboard  as  a  sight  example  will  think  that 
12  is  the  quotient  of  ^)24.  For  purposes  of  drill,  this  last  form 
should  occasionally  be  employed  in  sight  work,  as  should  be 

24 
the  third  form  of  division,  —  • 

2 

511.  Example  6 :  There  can  be  bought  as  many  bars  of  soap 
as  there  are  quarters  in  $3J,  or  13  bars.  Example  8 :  As  many 
yards  as  there  are  quarters  in  $5}.  Example  9  :  As  many  bushels 
as  there  are  quarters  in  $  10J . 

While  set  forms  of  analysis  should  not  be  required  in  any 
grades,  older  pupils  should  be  led  to  use  such  as  are  most  likely 
to  lead  to  an  intelligent  appreciation  of  mathematical  principles. 
From  the  beginning  of  about  the  fifth  year,  the  science  of  arith- 
metic should  begin  to  receive  some  attention,  but  not  so  much  as 
to  lessen  to  too  great  an  extent  the  time  that  should  be  devoted 
to  arithmetic  as  an  art. 

516.  These  examples  are  introduced  to  lead  up  to  division  of 
Federal  money.     From  their  previous  experience,  the  scholars 
will  readily  work  the  first  example,  for  instance,  by  changing 
it   to   the   form   $24£  H-  $£  =  *f-  --  £  =  49.     No.    2   becomes 
$12J  +  $J;     No.  3,  $26H-$J  =  ^-^;  etc. 

Without  laying  much  stress  upon  the  terms  "  abstract "  and 
"  concrete,"  the  teacher  should  bring  her  pupils  to  understand 
that  the  quotient  of  the  first  example  is  49,  —  not  49  dollars. 

517.  While  giving  the  answers  to  these  exercises,  the  children 
should  be  able  to  state,  after  proper  questioning,  that  the  divi- 
dend must  be  of  the  same  denomination  as  the  divisor.      In 
2  ft.  -f-8  in.,  instead  of  changing  the  divisor  to  -J  ft.,  they  natu- 
rally reduce  the  dividend  to  24  inches,  even  if  in  2  ft.  -*-  6  in. 
they  may  have  used  £  ft.  as  the  divisor. 


50  MANUAL   FOR  TEACHERS 

518.  Changing  the    dividend  to   cents,   No.   1    becomes   400 
cents  -f- 10  cents,  or  400  -*- 10.     To  No.  11,  many  will  give  50  as 
the  result,  unless  previously  well  taught.    In  No.  12,  the  denomi- 
nation of  both  terms  being  the  same,  the  problem  becomes  3  -f-  \, 
or  12  quarters  -*- 1  quarter,  rather  than  300  -=-  25.     Nos.  13-20 
are  more  readily  worked  by  reducing  each  divisor  to  a  fraction 
of  a  dollar.     Pupils  should  understand  that  the  answer  is  the 
same  whether  the  dividend  is  changed  to  the  same  denomination 
as  the  divisor,  or  vice  versa. 

519.  Some  teachers  write  this  example  .36)27.00.     It  will  be 
found  safer  to  make  the  terms  whole  numbers  by  changing  both 
to  cents. 

520.  The  method  suggested  for  the  first  example,  11000  -*-  275, 
is  the  more  general  one,  although  longer,  perhaps,  than  110-^-2}. 
Do  not  permit  long  division  in  No.  2.     In  No.  3,  after  writing 
14000  -j-  560,  the  pupils  should  strike  out  a  cipher   from  each 
term :  this  should  be  insisted  upon  whenever '  the  divisor  ends 
in  a  cipher.     In  No.  4,  either  74^-  -*-  %  or  7450  -5-  50  should  be 
accepted.     If  the  work  in  No.  7  takes  the  form  75)$  27.00,  the 
answer  should  be  $  .36  ;  if  the  pupil  writes  75)2700^,  his  answer 
should  be  36  cents.     The  first  form  is  the  one  employed  in  the 
work  of  preceding  chapters,  and  no  change  should  be  suggested. 

521.  These  drills  in  obtaining  approximate  results  are  intended 
to  lead  the  pupil  to  such  an  examination  of  his  answer  as  will 
prevent  his  being  satisfied  with  one  very  much  out  of  the  way. 
It  should  not  be  expected  that  the  same  approximation  will  be 
obtained  by  all  the  members  of  a  class. 

2.  4200-^-200.  6.  30  +  38. 

3.  Jx48.  7.  175-*- 26. 

4.  12000  -*-  2000.  8.  19  X  10. 

5.  $2x99,  or  $1.95x100,  9.  87-50. 

or  $2  x  100.  10.    5x5x5. 


NOTES  ON  CHAPTER  SIX  51 

526.  Formal  instruction  in  denominate  numbers  should  be 
deferred  for  a  year  or  more.  The  average  scholar  will  be  able 
to  solve  all  these  problems  if  left  to  himself. 

528-532.  See  notes  on  previous  drills  of  this  kind,  Arts.  286 
and  350.  Special  exercises  in  multiplication  are  regularly  given 
by  some  teachers  in  the  following  manner : 

2,  12,  22,  32,  42,  52,  62,  72,  82,  92 
X5 

A  horizontal  or  a  vertical  row  of  numbers  ending  in  2,  for 
instance,  is  written  on  the  board  with,  say,  5  as  a  multiplier. 
Attention  is  called  to  the  fact  that  2  X  5  is  10,  so  that  all  of 
these  products  must  end  in  0. 

The  pupils  are  also  reminded  that  when  the  multiplicand  is  a 
number  of  two  figures,  1  must  be  carried  to  the  product  of  5 
times  the  tens'  figure.  When  the  teacher  points  to  12,  the  pupil 
says  5,  6,  60  —  the  first  number  (5)  being  the  product  of  the 
multiplier  and  the  tens'  figure  of  the  multiplicand ;  the  second, 
(6)  being  this  product  increased  by  the  carrying  figure  1 ;  the 
third  being  the  result,  which  has  been  completed  by  annexing 
the  units'  figure  (0)  of  the  first  product  (2  X  5). 

Pointing  to  52,  the  pupil  says  25,  26,  260 ;  to  92,  he  says  45, 
46,  460.  After  sufficient  drill  with  5  as  a  multiplier,  it  is 
replaced  successively  by  6,  7,  8,  etc.  The  row  of  multiplicands 
is  also  changed  to  3,  13,  23,  etc. ;  4,  14,  24,  etc. ;  when  the 
previous  row  has  been  employed  with  all  of  the  multipliers, 
say  from  5  to  12.  With  sufficient  practice  of  this  kind,  pupils 
become  able  to  give  the  product  of  any  number  of  two  figures,  by 
multipliers  to  12,  with  great  readiness. 

Some  teachers,  however,  prefer  in  oral  multiplication  to  use 
the  method  previously  suggested  of  commencing  to  multiply  at 
the  tens'  figure  of  the  multiplicand.  In  finding  5  times  38,  the 
pupil  takes  the  latter  number  as  it  is  given,  thirty  —  eight,  and 
multiplies  in  the  same  order,  obtaining  150  and  40,  or  190. 


52  MANUAL    FOR   TEACHERS 

534-535.   Long-division  drills.     See  Arts.  321  and  397-401. 
538.   See  Art.  563,  p.  55,  and  Arithmetic,  Art.  385. 
540.   See  Art.  384. 

543.  In  problems  of  this  kind,  writing  the  given  numbers  in 
the  places  called  for  by  the  conditions  of  each  example  helps 
the  pupil  in  his  solution.  After  writing  No.  5,  as  here  indi- 

1.    68  2-  _L  5.       ? 

43  24)264  -89 

?  92 

150 

cated,  he  can  see  that  addition  is  the  required  operation  more 
readily  than  if  he  endeavors  to  determine  it  from  the  words  of 
the  problem. 

546.  Teachers  should  not  weary  pupils  by  giving  too  many 
items  in  the  earlier  bills.  It  is  useful  to  employ  occasionally 
such  quantities  and  prices  as  will  not  require  the  use  of  a  sepa- 
rate piece  of  paper  to  perform  the  necessary  multiplications. 
In  No.  1,  for  instance,  the  pupil  should  be  compelled  to  fill  out 
the  cost  of  each  item  without  recourse  to  his  slate.  If  he  does 
not  know  the  product  of  16  X  5,  he  should  multiply  one  figure 
at  a  time,  writing  the  result  in  its  proper  place.  Except,  pos- 
sibly, No.  2,  the  other  bills  called  for  under  this  section  should 
be  made  out  in  the  way  suggested  for  No.  1,  the  use  of  a  slate 
or  other  paper  not  being  permitted. 

The  form  given  in  the  text-book  is  the  one  generally  followed 
by  business  men.  The  first  two  vertical  lines  are  kept  to  enclose 
the  day  of  the  month  (see  Arithmetic,  Art.  642).  The  total 
cost  of  each  item  is  placed  in  the  first  columns  of  dollars  and 
cents,  the  amount  of  the  bill  being  placed  in  the  last  columns, 
and  on  the  line  below  the  last  item.  When  a  single  article  is 
sold,  its  cost  is  placed  directly  in  the  first  columns  of  dollars  and 


NOTES   ON  CHAPTER  SIX  53 

cents,  and  is  not  written  in  the  column  of  prices.  Unnecessary 
words,  —  at  or  @,  for  instance,  per  yd.,  lb.,  etc.,  —  are  never 
used ;  nor  are  commas  employed  after  the  names  of  the  articles. 
It  is  now  customary  to  omit  the  period  after  the  date  and  after 
the  name  of  the  seller.  The  names  of  the  articles  are  generally 
commenced  with  capitals,  and  the  quantities  are  written  with 
small  letters.  The  heading  given  is  the  one  most  frequently 
used,  though  other  forms  are  common ;  such  as 

ABRAHAM  AND  STRAUS 

Sold  to  MRS.  H.  T.  SHORT 

Pupils  should  write  the  cost  of  10 J  Ib.  of  chicken  @  30^,  in 
No.  4,  as  $3.08.  Fractions  of  cents  should  not  appear  in  the 
results;  those  below  \$  being  rejected,  and  those  of  \f  and 
higher  being  considered  \f. 

547-551.  When  pupils  have  become  familiar  with  the  nota- 
tion and  numeration  of  decimals,  the  remaining  decimal  work  of 
this  chapter  should  not  require  much  discussion. 

554.  After  working  Nos.  1-4,  pupils  should  be  left  to  them- 
selves to  arrange  No.  5.     Nearly  all  of  them  will  place  the 
numbers  in  their  proper  places.     Neither  require  nor  permit 
unnecessary  ciphers  to  be  employed  to  fill  out  all  the  numbers 
to  three  decimal  places. 

555.  The  above  suggestions  apply  here. 

559.  In  the  product  of  .36  by  3,  pupils  will  naturally  place 
the  decimal  point  where  it  belongs,  as  they  will  in  example 
No.  42.  Before  working  No.  43,  they  should  be  required  to 
deduce  the  rule  for  "pointing  off."  All  unnecessary  ciphers 
should  be  canceled  in  the  product,  the  answer  to  No.  52  being 
read  by  the  pupil  as  960,  not  960  and  no  tenths  —  960.0. 


54  MANUAL   FOR   TEACHERS 

560.  These  exercises   should   lead   scholars  to   see   that  the 
number  of  decimal   places   in  the   multiplicand    cancels  a  cor- 
responding number  of  ciphers  in  the  multiplier. 

561.  In  giving  the  quotient  of  No.  11,  a  pupil  may  write 
£>AV     If  called  upon  to  read  this  answer,  he  will  see  that  9.32 
expresses  the  same   result.      He   will   readily  understand   that 
yflhr  ig  a^so  written  .086.     After  a  few  examples,  he  can  state 
the  rule. 

From  this  point,  the  teacher  may  change  the  method  of 
dividing  by  a  number  ending  in  one  or  more  ciphers.  Instead 
of  marking  off,  by  a  line,  a  corresponding  number  of  figures 
from  the  right  of  the  dividend,  the  pupil  can  locate  the  decimal 
point  in  the  proper  place.  In  No.  17,  the  decimal  point  will 
be  moved  one  place  to  the  left,  to  divide  by  10;  moving  it 
two  places  to  the  left  in  the  dividend  of  No.  18  will  give 
its  quotient ;  etc. 

563.  The  rule  for  "  pointing  off"  should  be  deduced  from  the 
sight  exercises  of  Art.  562.  In  dividing  8  and  64  hundredths 
by  2,  the  pupil  will,  without  prompting,  obtain  4  and  32  hun- 
dredths. When  he  comes  to  No.  5,  8.4  -s-  5,  he  can  be  led  to 
see  that  this  example  is  the  equivalent  of  No.  4,  8.40  -*-  5,  in 
which  the  quotient  is  1.68.  Nos.  9  and  10  also  require  the 
annexation  of  a  cipher  at  the  right  of  the  dividend. 

When  the  scholars  understand  that  the  quotient  must  contain 
the  same  number  of  decimal  places  as  the  dividend,  including 
any  ciphers  that  may  have  been  annexed  to  the  latter,  they 
should  be  taught  the  method  a  business  man  would  employ. 
The  latter,  in  dividing  120  by  64,  does  not  find  it  necessary 
to  write  ciphers  in  the  dividend,  and  then  to  count  the  number 
thus  annexed,  in  order  to  determine  the  position  of  the  decimal 
point  in  the  quotient.  He  sees  at  once  that  the  result  is  1  and 
a  decimal,  and  he  places  the  point  after  the  1,  before  he  writes 
the  next  quotient  figure. 


NOTES  ON  CHAPTER  SIX  55 

In  a  short-division  example,  the  pupils  should  write  the 
decimal  point  in  the  quotient  when  they  reach  it  in  the  divi- 
dend, placing  it  under  the  latter.  In  long  division,  the  decimal 
point  in  the  quotient  is  placed  over  the  point  in  the  dividend. 
While  the  scholars  have  been  warned  in  their  early  work  in 
short  division  against  writing  02  as  the  quotient  of  8)16,  they 
will  see  the  need  of  the  prefixed  cipher  in  the  answer,  .02, 
to  8). 16.  From  the  inspection  of  a  few  examples  of  this  kind, 
they  will  understand  that  each  figure  of  the  dividend  after  the 
decimal  point  requires  a  quotient  figure  (or  cipher). 

In   the   long-division   examples   worked    out  in  1.875 

Art.  563  of  the  Arithmetic,  the  partial   products    64)120. 
are  omitted,  to  show  how  some  European  countries  560 

shorten  work  of  this  kind.      The  horizontal  lines  480 

given  in  the  text-book  are  not  used.  320 

After  the  pupil  writes  the  quotient  figure  1,  he  0 

subtracts  by  the  "  building-up  "  method.  The  second  remainder, 
48,  is  obtained  by  saying  8  fours  are  32,  and  8  (writing  it)  are 
40 ;  8  sixes  are  48,  and  4  (carried)  are  52,  and  4  (writing  it)  are 
56.  See  Arithmetic,  Art.  385. 


564.  Teachers  should  not  forget  that  systematic  instruction 
in  decimal  fractions  belongs  to  the  sixth  school  year.  They 
should  be  content  if  their  pupils  learn  to  place  correctly  the 
decimal  point  in  the  quotient. 

Nos.  21-30  should  be  considered  rather  as  examples  in  divi- 
sion, than  as  examples  in  the  reduction  of  common  fractions 
to  decimal  ones.  By  J  is  meant  1  -*-  4 ;  and  to  solve  it,  the 
pupil  may  write  4)1.00,  as  in  No.  11.  He  should  learn  by 
degrees,  however,  that  it  is  not  necessary  to  write  all  the  ciphers 
in  the  dividend  in  order  to  obtain  the  result.  The  answer  to 
No.  22  can  be  derived  by  a  bright  scholar  from  8)1.0  just  as  well 
as  from  8)1.000.  He  may  desire  the  first  cipher  as  a  starting- 
point,  but  he  finds  the  others  superfluous. 


56  MANUAL    FOR   TEACHERS 

565.  These  problems  contain  a  few  simple  applications  of 
the  decimal  work  learned  thus  far.  In  No.  1,  the  value  of  the 
franc  may  be  given  in  the  fractional  form  also,  19^^,  and  the 
problem  worked  fractionally  and  decimally. 

Approximate  sight  results  might  be  asked  before  written 
work  is  begun.  Taking  the  franc  as  about  20^,  or  $£,  the  pupil 
should  say  that  the  answer  to  No.  1  is  less  than  $250.  Assum- 
ing 40  inches,  or  1-J-  yd.  as  the  length  of  the  meter,  1800  meters 
would  be  equal  to  2000  yards.  No.  33  can  be  solved  without 
using  the  pencil.  In  No.  38,  the  first  result  may  be  in  the 
form  of  a  common  fraction,  %  peck,  to  be  changed,  as  in  No.  22, 
to  .125  peck. 

569.  The  remarks  made  in  Art.  564  as  to  the  formal  teaching 
of  decimal  fractions  is  equally  applicable  to  the  subject  of 
measurements.  At  one  time,  all  instruction  in  mensuration 
was  deferred  until  the  last  year  of  the  common-school  course. 
At  present,  this  subject  is  generally  taken  up  in  connection 
with  the  systematic  work  in  denominate  numbers ;  but  there 
is  no  good  reason  why  pupils  compelled  to  leave  school  by  the 
end  of  the  fifth  year  of  the  course  should  not  receive  so  much 
practice  in  finding  the  areas  of  rectangles  as  they  have  time 
for  and  can  readily  understand. 

In  most  city  schools,  the  children  of  this  grade  know  from 
their  lessons  in  form  and  drawing  what  is  meant  by  a  square 
and  a  rectangle.  If  pupils  are  not  familiar  with  these  terms, 
they  should  be  explained. 

That  scholars  may  obtain  a  good  idea  of  a  square  inch,  they 
should  be  required  to  cut  out  a  number  of  square  pieces  of 
paper,  each  side  measuring  an  inch.  These  squares  should  then 
be  used  in  determining  the  number  of  square  inches  in  the  two 
rectangles  next  drawn ;  2  inches  by  1  inch,  and  3  inches  by  2 
inches.  Children  that  determine  the  areas  of  these  rectangles 
by  covering  them  with  their  paper  squares  will  have  a  better 
knowledge  of  2  square  inches  and  6  square  inches  than  if  they 


NOTES   ON   CHAPTER   SIX  57 

merely  divide  the  rectangles  by  lines  as  suggested  in  the  text- 
book. 

The  larger  rectangles,  6x3  and  4x4,  may  be  cut  up  into 
inch  squares  by  drawing  lines,  and  the  rule  for  obtaining  the 
area  of  each  deduced  from  an  examination  of  the  figures.  In 
the  first,  the  pupil  will  see  that  he  has  3  rows  of  squares,  6  to  a 
row  (or  6  rows,  3  to  a  row),  making  6  X  3,  or  18,  squares.  The 
rule  should  take  in  his  mind  some  such  form  as  this :  the  num- 
ber of  square  inches  in  a  rectangle  is  equal  to  the  product  of  the 
number  of  inches  in  one  dimension  multiplied  by  the  number  of 
inches  in  the  other;  but  he  should  not  be  required  to  give  it 
expression.  The  teacher  should  take  care  that  he  does  not  think 
that  "  inches  by  inches  give  square  inches." 

570.  As  has  already  been  said,  the  chief  use,  in  arithmetical 
instruction,  of  objects,  diagrams,  etc.,  is  to  enable  pupils  to  work 
without  them.  After  the  scholars  understand  how  to  obtain  the 
area  of  a  rectangle,  they  should  cease  to  draw  the  figure  and  to 
subdivide  it  into  squares. 

It  will  be  noted  that  the  answers  to  the  first  20  examples  are 
to  be  given  in  square  inches.  In  Nos.  11-20  each  dimension 
should  be  reduced  to  inches  before  the  multiplication  is  per- 
formed. 


NOTES   ON   CHAPTER  SEVEN 

At  this  point  regular  fraction  work  should  begin.  From  time 
to  time,  as  occasion  offers,  the  meanings  of  the  technical  terms 
should  be  elicited  from  the  pupils  ;  but  the  teacher  should  neither 
accept  a  memorized  definition  that  is  not  thoroughly  understood, 
nor  should  she  require  absolute  correctness  in  the  phraseology  of 
a  definition  made  by  a  scholar. 

577.  While  the  denominators  of  the  fractions  should  generally 
be  small ;  and  while  common  denominators  should,  as  a  rule,  be 
determinable  by  inspection,  it  is  necessary,  nevertheless,  that  the 
children  be  taught  how  to  handle  such  other  fractions  as  they 
may  occasionally  meet.  It  is  not  necessary  that  they  should 
grasp  the  exact  meaning  of  |-|J  in  the  answer  to  No.  4,  although 
proper  teaching  may  enable  them  to  see  later  on  that  this  fraction 
approximates  ££#,  f . 

In  these  earlier  examples,  the  inspection  method  of  determining 
the  common  denominator  is  continued. 

580.  Besides  being  necessary  as  a  preliminary  to  subsequent 
work  in  fractions,  expertness  in  determining  the  factors  of  a  num- 
ber is  useful  in  enabling  pupils  to  shorten  their  work  by  cancel- 
lation.    The  teacher  should  use  these  and  similar  exercises  again 
and  again,  for  a  few  minutes  at  a  time,  until  her  scholars  can 
give  the  answers  with  great  rapidity. 

581.  The  pupils  will  need  to  learn  the  difference  between  the 
three  factors  of  12,  and  three  divisors  of  12.     The  factors  will  be 
2,  2,  and  3,  because  their  product,  2x2x3  equals  12.     The 
divisors  of  12  are  2,  3,  4,  and  6. 

58 


NOTES  ON   CHAPTfclW&vl#:MITY  59 


These  exercises,  as  well  as  those  in  ArlS.™  bSU-o85,  are  not 
so  valuable  as  to  demand  the  reviews  suggested  for  those  in 
Art.  580. 

In  finding  three  (or  more)  factors  of  a  number,  the  scholar 
should  commence  with  the  smallest.  The  first  of  the  three 
factors  of  8  is  2  ;  dividing  8  by  this  factor,  4  is  obtained,  of 
which  2  and  2  are  the  factors. 

The  three  factors  of  18  are  2  X  3  X  3  ;  of  20,  are  2  X  2  X  5  ; 
of  27,  3  X  3  X  3  ;  etc.  ;  etc. 

582.  It  is  customary  to  define  a  prime  number  as  one  that 
has  no  factor  except  itself  and  unity.  The  omission  of  the  last 
four  words  will  not  mislead  any  person,  as  there  could  be  no 
prime  numbers  if  1  were  considered  a  factor.  When  the  factors 
of  a  number,  say  20,  are  asked  for,  no  one  gives  1  X  1  X  1  X  1  X 
Ix2x2x5as  the  answer,  or  says  that  20  has  eight  (or  more) 
factors. 


In  reducing  these  fractions  to  lowest  terms,  it  is  not  nec- 
essary that  the  pupils  should  use  the  greatest  common  divisor. 
See  Arithmetic,  Art.  592.  On  the  other  hand,  they  should  not 
waste  time  in  dividing  each  term  by  5,  if  25  is  a  common  divisor. 

589.  Pupils  should  not  be  permitted  to  forget  these  tests  of 
the  divisibility  of  numbers.  To  those  given  in  the  text-book, 
there  may  be  added  that  when  a  number  divisible  by  3  is  even, 
it  is  also  divisible  by  6. 

While  a  teacher  should  know  that  1001,  with,  of  course,  its 
multiples,  — 2002,  6006,  15015,  etc.,  — is  divisible  by  7,  11,  and 
13,  she  should  not  burden  her  scholars  with  the  information ; 
nor  should  she  dwell  upon  the  test  of  divisibility  by  8. 

591.  Beginners  should  be  taught  only  one  method  of  finding 
the  greatest  common  divisor,  and  the  one  here  given  is  applicable 
to  all  kinds  of  numbers.  Teachers  should  not  bewilder  young 
pupils  by  endeavoring  to  make  them  understand  the  principles 
upon  which  this  method  is  based. 


60  MANUAL    FOE   TEACHERS 

595.    Many  teachers  prefer  to  permit  their  pupils  to  write 
-r/itfi/toio     down  all  of  the  denominators.  kiK1. 

-JLflr-p-L4^-Jr-LZ  ,->.- 

— « «     then  to  strike  out  any  one  that  is 

repeated  or  that  is  the  factor  of  any 
other.     They  think  the  pupil  is  less 

likely  to  make  a  mistake  by  following  this  plan.  In  no  case 
should  scholars  be  permitted  to  begin  work  before  rejecting  or 
striking  out  the  unnecessary  numbers. 

605-606.  Pupils  that  have  had  regular  drills  in  the  com- 
binations given  in  the  previous  chapters  will  be  able  to  take 
the  extra  step  required  by  these  examples.  See  Arts.  286-290 
and  350-352. 

607.  A  scholar  that  can  find  mentally  the  cost  of  47  articles 
at  25  ^  each  should  be  able  to  give  the  product  of  47  X  25  or 
25  X  36  without  using  the  pencil,  and  the  teacher  should  give 
him  a  chance  to  determine  for  himself  the  method  of  doing  it. 

609.  Such  questions  as  18f  -*-  2J  can  be  worked  by  the  method 
given  in  the  last  chapter ;  viz.,  56  thirds  •*•  8  thirds  —  56  -f-  8  =  7. 
Those  contained  in  the  4th  column  should  not  be  used  until  the 
pupils  have  had  formal  instruction  in  division  of  fractions.     When 
they  are  taken  up,  the  method  followed  should  be  that  given 
above,  the  fractions  being  reduced  to  a  common  denominator,  etc. 
i.  -*.  I  =  3  sixths  ni-  4  sixths  =  3  -s-  4  =  £ ;  $-*-£  =  8  twelfths  -*-  9 
twelfths  =  | ;  etc.     See  Arithmetic,  Art.  639,  note. 

610.  No.  9 :  For  $1.25  I  can  buy  5  times  as  many  pounds  as 
for  25^,  or  15  pounds.     No.  16:  For  18^  there  can  be  bought 
|f  lb.,  or  f  lb.,  or  12  oz. 

613.  In  giving  answers  to  these  exercises,  pupils  should  be 
permitted  to  write  the  fraction  first  and  then  the  whole  number. 

The  object  of  these  exercises  is  to  accustom  the  scholars  to  dis- 
pense with  writing  unnecessary  reductions  in  adding  and  sub- 
tracting simple  fractions. 


NOTES  ON  CHAPTER  SEVEN  61 

614.  Another  method  of  finding  the  difference  between  llf 
0}  is  to  take  6£  from  7,  obtaining  £,  and  to  add  to  this  the 
difference  between  7  and  llf,  or  4f. 

In  some  classes  of  sight  exercises,  those  given  in  Arts.  587, 
594,  and  605-609,  for  instance,  the  pupils  should  not  take  pens 
to  write  the  result  until  told  to  do  so  by  the  teacher,  after 
sufficient  time  has  been  given  to  obtain  the  answer.  In  the 
exercises  of  Arts.  613  and  614,  the  pupils  should  be  permitted 
to  take  their  pens  at  once,  and  to  write  each  part  of  the  result  as 
soon  as  it  has  been  obtained.  Arts.  584,  650-654,  699,  etc.,  also 
contain  exercises  of  this  kind. 

616.   See  Art.  563.     The  first  quotient  figure,  49544 

4,  is  written.     The  pupil  then  says  4  sixes  are  24      36)17837 
an&  4  (writing  it)  are  28 ;  4  threes  are  12,  and  2  343 

(to  carry)  are  14,  and  3  (writing  it)  are  17.     This  ^97 

gives  the  first  remainder,  34.     The  next  figure,  3,  -^ 

is  then  brought  down,  and  9  is  written  in  the 
quotient.     The  product  of  9  times  36  is  subtracted  from  343  as 
given  above,  to  obtain  the  next  remainder,  19.     The  pupil  says 
9  sixes  are  54  and  9  (writing  it)  are  63 ;  9  threes  are  27  and  6 
(to  carry)  are  33,  and  1  (writing  it)  are  34. 

618.  While  pupils  should   be  encouraged   to   shorten   their 
work  by  cancellation,  the  slower  children  should  not  be  censured 
when  they  overlook  some  cases  in  which  it  is  possible  to  employ 
this  expedient.     In  these  examples,  however,  all  the  scholars 
should  be  required  to  indicate  the  operations,  and  then  to  cancel. 

619.  It  is  not  supposed  that  pupils  should  write  answers  to 
these  questions,  as  is  generally  done  in  the  case  of  the  oral 
problems.     These  exercises  are  intended  to  lead  up  to  the  rule 
for  multiplication  of  fractions.     Diagrams  should  be  drawn  on 
the  blackboard  by  the  pupils,  to  illustrate  the  answers,  but  the 
teacher  should  refrain  as  much  as  possible  from  "  explaining." 


62  MANUAL    FOR   TEACHEBS 

The  board  work  should  be  done  chiefly  by  the  more  backward 
members  of  the  class  rather  than  by  the  brighter  ones.  In 
illustrating  fractions  by  diagrams,  the  unit  employed  should 
generally  be  a  circle,  the  part  dealt  with  being  distinguished  by 
shading. 

The  zealous  teacher  should  not  become  discouraged  at  the 
inability  of  some  members  of  the  class  to  thoroughly  grasp 
the  mathematical  principles  involved  in  this  and  other  operations. 
Even  the  ability  to  handle  fractions  mechanically  will  be  of 
great  use  in  after  life,  and  all  the  pupils  can  be  taught  at  least 
this  much. 

624.  No.  31  reduces  to  9f  -5-  3,  which  can  readily  be  worked 
by  the  pupils  without  assistance.      No.   34,  40|-  -4-  8,  may  be 
difficult  for  some ;  but  the  teacher  should  not  offer  help  too  soon. 
The  second  term  of  No.  36  is  easily  obtained;  and  No.  40  will 
give  no  trouble. 

625.  These  exercises  are  to  be  used  in  the  same  way  as  those 
in  Art.  619. 

Many  children  find  it  difficult  to  understand  that  4  fourths  -5- 
3  fourths  =  1^.  They  think  that  the  answer  should  be  1^,  reason- 
ing it  out  in  some  such  way  as  this :  3  fourths  into  4  fourths  goes 
1  time  and  1  fourth  over.  They  fail  to  recollect  that  the 
remainder  in  division  is  written  over  the  divisor,  which  would 
give  them  1  jiSb,  or  1^.  If  they  have  been  well  taught  previously, 
they  may  remember  that  4  fourths  -4-  3  fourths  —  4  -4-  3  =  1£. 
Even  a  fairly  bright  pupil,  when  asked  how  much  tea  at  $f 
per  Ib.  can  be  bought  for  $1,  will  sometimes  reply,  "A  pound 
and  a  quarter."  When  he  is  told  that  his  answer  is  correct  if  he 
means  by  it  a  pound  of  tea  and  a  silver  quarter,  he  sees  the  mis- 
take and  changes  the  result  to  "  a  pound  and  a  third." 

626.  If  1-4--J  is  made  concrete,  a  pupil  can  more  easily  show 
by  a  diagram  that  the  result  is  1£.     A  problem  of  this  kind  may 


NOTES  ON    CHAPTER    SEVEN  63 

be  given  :  If  it  requires  J  yd.  of  material  for  an  apron,  how 
many  aprons  can  be  made  from  1  yard  ?  A  rectangle  is  drawn 
to  represent  the  yard  of  material,  and  it  is  divided  into  thirds. 
Underneath,  a  rectangle  two-thirds  as  long  is  drawn  to  represent 
the  quantity  required  for  an  apron.  When  the  pupil  compares 
the  two  rectangles,  he  sees  that  the  portion  remaining  after  one 
apron  is  made  will  supply  sufficient  material  for  one-half  of 
another. 

627.  While  it  is  generally  better  in  oral  work  to  divide  one 
fraction  by  another  by  reducing  both  to  a  common  denominator, 
it  will  be  found  simpler  in  written  work  to  have  pupils  invert 
the  divisor. 

631.  "  Invert  the  divisor,  and  proceed  as  in  multiplication," 
is  the  rule  generally  followed. 

634.  No.  33  can  be  shortened  by  writing  it  as  follows,  before 
beginning  work  :  *f-  x  J  X  $,  the  divisor  being  inverted.  No.  34 
should  be  treated  in  the  same  way  :  (20  -*-  J)  X  f  =  *£•  X  f  X  }.  The 
divisor  of  No.  35  consists  of  two  fractions,  both  of  which  should 
be  inverted  :  20  -*-  (£  X  f  )  =  *£  X  f  X  $.  This  method  should  be 
followed  with  Nos.  36,  41,  and  42.  The  first  of  these  becomes 
y  X  f  X  |  ;  the  next,  *j-x$x*£  x&x$>  an(*  tne  next> 


635.  Each  teacher  must  determine  for  herself  what  method 
of  analysis  should  be  encouraged  in  such  questions  as  Nos.  4,  8, 
13,  and  14.  While  set  forms  should  be  avoided,  children  need 
direction  in  the  solution  of  problems  of  this  kind. 

In  solving  No.  4,  for  instance,  the  greater  number  of  teachers 
prefer  to  have  pupils  first  find  the  cost  of  -J  yd.  When  this 
method  is  followed,  care  must  be  taken  that  all  the  pupils  under- 
stand why  J  yd.  costs  one-half  of  20  cents.  This  may  be  made 
clearer  to  some  by  writing  the  fractions  in  this  way  :  If  2  thirds 


64  MANUAL    FOR   TEACHERS 

(or  parts)  cost  20  cents,  what  will  1  third  (or  part)  cost?  A 
diagram  similar  to  that  given  in  Arithmetic,  Art.  636,  may  help 
others  to  understand  the  method. 

Other  successful  teachers  think  the  written  work  is  benefited 
by  treating  these  examples  as  problems  in  division.  They  lead 
their  children  to  determine  in  each  case  what  operation  is  in- 
volved, by  requiring  them  to  consider  what  they  would  do  if  the 
fraction  were  a  whole  number.  In  No.  1,  for  example,  the  cost 
of  16  balls  at  $  3  each  would  be  $  3  X  16.  In  No.  2,  the  pupil 
would  say,  "If  I  paid  $  12  for  base-balls  at  $3  each,  the  number 
of  balls  would  equal  12-^3.  I  must,  therefore,  divide."  He 
mentally  inverts  the  divisor,  -|,  then  cancels,  etc. 

636.  The  scholars  should  be  allowed  sufficient  time  to  work 
these  out  in  their  own  way. 

639.  No.  4:  24£  -«-  3J  =  -4/  -*-  %  =  49  -*-  7.  Some  pupils  will 
see  that  time  is  lost  in  No.  6  by  finding  the  cost  of  a  pound. 
No.  7  is  an  example  in  division:  l-f-*-2J-  =  -f-5--J  =  -JX-§-;  or 
V1  -H  Jg§-  =  10  -4- 15,  etc.  In  No.  16,  36  hats  will  cost  3  times  $  7. 

642.   See  Art.  546. 

649.  In  multiplying  by   25,  the  pupil  is  generally  told  to 
annex  two  ciphers  and  to  divide  by  4.     In  mental  work  espe- 
cially, the  annexation  of  the  ciphers  confuses  some  scholars  by 
giving  them  a  larger  dividend  than  is  really  required.     The 
product  of  25  times  19  may  be  obtained  more  easily  by  taking 
one-fourth  of  19,  or  4f ,  and  changing  this  quotient  to  475,  than 
by  finding  one-fourth  of  1900.     In  No.  9,  the  pupil  should  see 
that  at  $100  per  bbl.,  the  pork  would  cost  $5600,  and  that  at 
$  12.50  per  bbl.  (£  of  $  100),  it  would  cost  |  of  $  5600. 

650.  In  No.  1,  divide  837  by  4,  and  for  the  1  remainder  af- 
fix 25  to  the  quotient.     In  No.  4,  annex  two  ciphers  to  the  quo- 
tient of  508  by  4.     In  No.  9,  affix  250  to  the  quotient  of  837  -*-  4. 


NOTES  ON   CHAPTER  SEVEN  65 

In  multiplying  6281  by  12J-,  No.  18,  divide  6281  by  8,  obtaining 
785,  and  annex  12£  for  the  1  remainder,  making  the  result 
78,512f 

654.  When  the  divisor  is  a  whole  number,  time  should  not  be 
wasted  in  changing  a  mixed  number  dividend  to  an  improper 
fraction.  Nos.  64,  65,  and  66  resemble  those  already  worked. 
In  No.  69,  after  obtaining  the  quotient  14,  there  will  be  a  re- 
mainder 2J-,  which  is  changed  to  f  and  divided  by  5,  giving  £  as 
the  result.  In  No.  69,  the  remainder,  5J,  is  changed  to  ^-, 
which  gives  |^  when  it  is  divided  by  8. 

656.  Some  mistakes  would  be  avoifled  if  pupils  would  learn 
to  ask  themselves  if  the  answer  they  have  obtained  is  a  reason- 
able one.  Permit  the  scholars  to  work  out  all  these  examples 
without  giving  them  a  rule  for  "  pointing  off." 

669.   See  Art.  521. 

3.  6x6.  7.  800-^100. 

4.  300-^12.  8.  8x8. 

5.  86x1.  9.  7x11. 

6.  36  -*-  4.  10.  64  xf. 

670. 

3.  25x12.  7.  800  x.l. 

4.  36-^6.  8.  8x8. 

5.  86-^-1.  9.  7x  12. 

6.  32x5.  10.  64  -H}. 

671. 

2.  $2|X200.  6.  25^X800. 

3.  25^x4.  7.  $2Jx20. 

4.  $  12  X  400.  8.  60^  X  1000. 

5.  $2x8.  9.  $5000  X  7. 

10.  $  1  X  6. 


bb  MANUAL   FOR   TEACHERS 

677.  Teachers    should    carefully    avoid    giving    unnecessary 
"  rules."     There  is  no  good  reason  why  an  average  pupil  should 
not  be  able  to  determine  for  himself  how  to  ascertain  what  part 
of  $15  a  man  has  spent  when  he  has  spent  $5.     While-  the  in- 
troduction of  fractions  into  such  an  example  makes  it  more  diffi- 
cult for  the  scholar  to  give  the  answer  off-hand,  his  instruction 
up  to  this  time  should  have  taught  him  that  the  same  process  is 
to  be  employed.     A  pupil  should  be  required  to  depend  upon 
himself  to  at  least  a  reasonable  extent. 

678.  As  a  preliminary  to  the  work  in  denominate  numbers  in 
the  next  three  pages,  the  teacher  should  place  on  the  board 
a  few  such  examples  as*  the  following,  to  which  the  scholars 
should  give  answers  at  sight : 

1£  qt.  1  qt.  1  pt.  3  qt.  1  qt.  1  pt. 

+ 1  j.  qt.  + 1  qt.  1  pt.  - 1  qt.  1  pt.  X2 

2)3  qt.  1  qt.  1  pt.)3  qt. 

Nearly  every  member  of  the  class  will  be  able  to  obtain  the 
results  in  a  moment,  without  any  suggestions  from  the  teacher. 
If  the  examples  are  left  on  the  board,  the  pupils  can  refer  to 
them  for  aid  in  working  some  of  those  found  in  the  text-book. 

The  teacher  that  wishes  to  develop  power  in  her  scholars 
should  be  careful  not  to  give  a  particle  more  assistance  than  is 
necessary.  She  should  permit  the  children  to  deduce  from  the 
above  examples  the  rules  necessary  to  solve  the  others,  being 
patient  if  the  pupils  are  somewhat  slow  in  doing  this  work. 
When,  however,  a  circuitous  method  has  been  employed,  she 
should  lead  the  class  to  see  how  the  work  can  be  improved  by 
the  use  of  a  shorter  way. 

680.  It  may  be  necessary  to  take  up  again,  for  purposes  of 
review,  the  preliminary  exercises  of  the  previous  chapter.  See 
Art.  569,  pp.  56  and  57. 


/  /»* 

NOTES  ON   CHAIVETl  SEVEN       ^VK/  67 

igfLCALlFOg^ 

681.  As  the  table  of  square  measure  is  not  introduced  until 
the  next  chapter,  it  will  be  necessary  to  reduce  to  yards  the 
dimensions  that  are  given  in  feet  or  inches. 

2.  18  yd.  by  21  yd.  8.   18  yd.  by  2  yd. 

3.  2  yd.  by  3  yd.  9.    16  yd.  by  15  yd. 
7.   9  yd.  by  32  yd.                   10.    1£  yd.  by  24  yd. 

682.  No.  14.    14  yd.  by  f  yd. 

No.  15.    8  pieces,  each  36  yd.  long  and  f-£  yd.,  or  {•  yd.  wide. 

No.  18.  See  Arithmetic,  Art.  818,  problem  20.  A  modifica- 
tion of  this  diagram,  showing  four  squares  instead  of  four  rec- 
tangles will  be  the  drawing  required,  except  that  the  squares 
above  and  below  need  not  necessarily  occupy  the  positions  there 
indicated. 


XI 

NOTES  ON  CHAPTER  EIGHT 

With  this  chapter  begins  the  regular  work  in  decimal  frac- 
tions, and  the  pupils  should  now  be  taught  the  principles  under- 
lying the  various  operations. 

685.  While  pupils  may  know  that  -2^-  means  that  23  is  to  be 
divided  by  8,  it  may  be  well  to  lead  them  again  to  see  that 
f  is  the  same  as  3  -5-  4,  or  \  of  3.     After  they  understand  that 
every  common  fraction  may  be  considered  an  "  indicated  division," 
they  will  understand  that  the  decimal  fraction  obtained  by  per- 
forming this  operation  is  the  equivalent  of  the  common  fraction 
whose  denominator  is  used  as  a  divisor  and  whose  numerator  is 
used  as  a  dividend.     See  Arts.  563  and  564. 

686.  As  previous  work  in  decimals  has  been  confined  chiefly 
to  three  places,  some  review  and  extension  of  the  notation  and 
numeration  exercises  of  Arts.  547-551  may  be  necessary. 

687.  After  writing  each  of  these  decimals  in  the  form  of  a 
common  fraction,  a  scholar  should  be  able  to  determine  at  a  glance 
whether  or  not  it  can  be  reduced  to  lower  terms.     This  reduction 
is  possible  when  the  decimal  is  an  even  number  or  terminates  in 
a  5. 

While  it  is  inadvisable  to  waste  time  in  calculating  the  great- 
est common  divisor,  pupils  should  be  encouraged  to  use  large 
divisors ;  4  rather  than  2,  when  possible,  and  25  rather  than  5. 

688.  The  common  fractions  contained  in  these  exercises  are 
such  as  do  not  require  much  calculating  to  change  them  to  deci- 


NOTES   ON  CHAPTER   EIGHT  69 

mals.  The  scholars  should  be  able  to  write  the  numbers  in 
vertical  columns  directly  from  the  text-book,  making  the  neces- 
sary reductions  mentally. 

In  reducing  ^  to  a  decimal,  it  may  be  easier  for  some  to  con- 
sider it  ^  of  J,  or  J  of  .25.  The  reduction  of  £J  is  simplified  by 
multiplying  each  term  by  2,  making  it  -j^,  or  .46,  instead  of 
dividing  23  by  50,  etc.,  etc. 

690.  Nos.  62,  64,  66,  and  68  may  be  worked  by  using  the 
common  fraction  given,  and  also  by  reducing  this  to  a  decimal 
before  performing  the  multiplication. 

See  Art.  563,  p.  55,  and  Art.  616. 

The  teacher  should  not  permit  the  employment  of  long 
division  in  these  examples.  In  No.  92,  the  children  can  see  that 
changing  the  dividend  to  .18756  divides  it  by  100,  and  that 
.18756  -+-  3  is  the  same  as  18.756  by  300.  See  Arithmetic,  Art. 
668. 

694.  Ciphers  at  the  right  of  a  decimal  should  be  rejected, 
excepting,  perhaps,  the  final  0  in  cents.  See  Nos.  3,  4,  and  10. 

2.  $.95x7.6.  6.  $22x108.745. 

3.  $2.80x48.6.  7.  $.75x148.6. 

4.  $21.30x39.25.  8.  $.13J  x  (2376^  12). 

5.  $.68x18.75.  9.  $35x4.5. 

10.   $  13.50  x  [(28  x  12)  H- 144]. 

While  it  is  inadvisable  to  confuse  children  by  too  many  short 
methods  in  the  earlier  stages,  they  should  be  encouraged  in  ex- 
amples like  the  foregoing  to  use  as  a  multiplier  the  number  that 
will  make  the  work  easier,  and  to  employ  a  common  fraction 
instead  of  a  decimal  whenever  the  use  of  the  former  would 
lighten  their  labor.  In  No.  1,  for  instance,  the  result  is  obtained 
with  fewer  figures  by  multiplying  24.4  by  6J,  instead  of 
6.25  X  24.4. 


70  MANUAL    FOR   TEACHERS 

695.   The  operation  should  first  be  indicated. 

$  .90  x  38648  $  .36  x  48576 

~60~  ~32~ 

$  5  X  18964  1 1.83  x  69104 

2000  2  X  56 

etc.,  etc. 

703-704.   See  notes  on  previous  special  drills,  Arts.  286  and 
350. 

705.  See  Arts.  528  and  649.     In  multiplying  46  by  3%  divide 
46  by  3,  whiclf  gives  15  J,  and  substitute  33^  for  the  fraction  in 
the  quotient,  thus  obtaining  the  result,  1533J. 

706.  975  *  25  =  9f  H-  J  =  9f  x  4.       433J  -*-  33£  =  4£  -*•  J- 
=  41  x  3. 

708.  Use  first  as  "  sight "  problems,  if  the  pupils  find  the 
numbers  too  large  to  be  carried  in  the  mind.     By  degrees,  how- 
ever, they  should  acquire  the  power  to  solve  problems  of  this 
kind  without  seeing  the  figures,  especially  when  the  operations 
are  not  numerous  or  involved. 

709.  In  such  examples  as  Nos.  1,  9,  10,  11,  and  the  like,  many 
children  fail  to  comprehend  the  form  of  analysis  generally  given. 
While  they  get  some  facility  in  applying  the  method,  they  do 
not  understand  the  underlying  principle.     In  finding  a  number, 
|-  of  which  is  180,  they  learn  to  divide  by  5  and  to  multiply  the 
quotient  by  6,  and  to  repeat  the  customary  formula,  without 
knowing  the  reasons  for  the  different  operations.     There  are  only 
four  fundamental  processes  in  arithmetic,  and  children  should 
be  taught  to  determine  for  themselves  which  to  use  in  a  given 
example  that  is  within  their  experience,  rather  than  to  depend 
upon  a  rule  which  they  do  not  fully  understand,  and  which  they 
are  likely  to  forget  or  to  misapply.     See  Art.  635.     A  few  dia- 
grams are  here  introduced,  to  be  used  by  the  teacher  that  does 


NOTES  ON   CHAPTER  EIGHT  71 

not  wish  her  pupils  to  obtain  in  No.  1,  for  instance,  the  length 
of  the  room  by  dividing  15  by  $.  The  five  spaces  in  the  width 
are  each  3  ft.,  which  will  make  the  length  18  ft.  When  a  scholar 
understands  this  from  the  diagram,  he  can  understand  that  when 
f  of  a  number  is  15,  £  is  15  -*-  5,  or  3. 

While,  for  purposes  of  drill,  many  "  abstract  "  examples  of  the 
same  kind  are  brought  together  in  one  place,  care  has  been  taken 
in  the  problems  to  avoid  having  two  consecutive  ones  alike  in 
character.  Problem  work  to  be  of  value  should  not  be  permitted 
to  become  mechanical.  Pupils  should  need  to  study  each  prob- 
lem to  determine  the  method  of  solving  it. 

710.  See  Arithmetic,  Art.  384. 

716.  By  placing  the  multiplier  at  the  right  of  the  multi- 
plicand, the  pupil  can  use  the  latter  as  the  first  partial  product, 
instead  of  writing  it  again,  as  he  would  be  compelled  to  do  if  the 
multiplier  were  placed  in  its  usual  position. 


717.   Some  teachers  might  prefer  to  place  the  product  by  8 
nd,  as  being  the  form  to 
^ichthescholarsaremore 


above  the  multiplicand,  as  being  the  form  to 


2304* 14  x  21  accustomed ; but  in  such  an 

example    as   No.   26,   the 
16123  latter  part  of  the  work  can  be  shortened 

169344  Ans.  on^v  bv  plac*ng  ^e  product  by  the  units' 

figure  under  the  other.  48600 

No.  28  should  be  worked  as  is  here  shown. 
Annexing  two  ciphers  to  the  multiplicand  and         3)97200 

multiplying  by  -J  gives  the  product  by  66$. 

^SsOoUU 

2300400 

719-720.  See  Art.  521. 

1.  24  Ib.  @$f  3.    64yd. 

2.  24  horses®  $125.  4.   485  bu. 


72  MANUAL   FOE  TEACHERS 

5.  96  Ib.  @$f  18.  60n-f 

6.  840yd.  @  $^.  19.  64-*-f 

7.  360yd.  @  $f  20.  28  ^  If . 

8.  48  cwt.  @  |f.  21.  17  X  4. 

9.  92  hats  @  $11  22.  256  X  f 

10.  128  Ib.  @  $f  23.  25  X  16. 

11.  27-s-f  24.  6x6. 

12.  300-s- If  25.  86  x  1. 

13.  24-j-f  26.  33x5. 

14.  15-*-f  27.  800  xf 

15.  60-*-2f  28.  8X8. 

16.  32-*-f  29.  7x11. 

17.  70-H-J.  30.  64  xf. 

721.  The  decimals  should  be  reduced  to  common  fractions 
whenever  the  work  is  rendered  easier  by  the  change. 

1.  360  xf  7.    72  xf  13.    84  X  f 

2.  560  X  f  8.    84  X  f .  14.    15  X  6. 

3.  240  xf  9-    96  X^.  15-      4  X  4. 

etc.  etc.  etc. 

722.  The  pupil  should  employ  such  method  as  is  best  adapted 
to  the  particular  example  : 

1.  240  -f-f  9.    48-^^.  17.      65-s-f 

2.  360 -*-f  10.    72-^.  18.    840-^-8. 

3.  45  ^f        '   .        11.    92000^2.  19.      11  -f-Ty 

etc.  etc.  etc. 

723.  Nos.  1  to  8  are  intended  to  furnish  practice  in  sight 
cancellation.     In  Nos.  13  to  16,  the  reduction  of  the  multiplier 
to  an  improper  fraction  will  simplify  the  work  for  some  pupils. 

726.   Whenever  possible,  the  least  common  denominator  should 
be  determined  by  inspection. 

735.   Do  not  give  "  rules."     See  Art.  678. 


NOTES  ON   CHAPTER  EIGHT  73 

736-737.  These  exercises  are  introduced  to  accustom  the 
pupils  to  add  and  to  subtract  simple  mixed  numbers  without 
rewriting  the  fractions  reduced  to  a  common  denominator. 

740.  In  these  and  other  similar  examples,  the  teacher  should 
not  anticipate  the  work  of  the  higher  grades  by  systematic  instruc- 
tion in  advanced  topics.  All  that  should  be  done  with  respect 
to  these  problems  is  to  show  the  pupils  that,  when  a  solution 
involves  multiplication  and  division,  time  may  frequently  be  saved 
by  means  of  cancellation.  The  pupils  should  be  permitted  to 
work  out  No.  1  at  length,  if  they  wish ;  after  which  they  should 
be  required  to  indicate  the  work  by  signs,  and  then  to  cancel. 
Division  should,  of  course,  be  indicated  by  writing  the  divisor 
as  a  denominator. 

Some  excellent  teachers  require  their  scholars  before  beginning 
work  on  a  problem  to  indicate  by  signs  all  the  operations  neces- 
sary to  its  solution,  thereby  compelling  them  to  study  the  con- 
ditions thoroughly  at  the  outset.  Too  many  pupils  commence  to 
add,  subtract,  etc.,  without  fully  realizing  what  is  required  in  a 
given  example. 

742-744.  See  Art.  678. 

745.  The  pupils  should  write  the  dimensions  on  each  diagram, 
changing  them,  when  necessary,  to  the  denomination  required  in 
the  answer. 

746.  The  formal  study  of  percentage  belongs  to  the  next  year 
of  the  course,  and  teachers  should  not  dwell  too  much  on  this 
topic.     After  the  pupils  understand  the  meaning  of  the  term  per 
cent,  they  should  be  able  to  work  the  examples  given.     Other 
technical   terms,   definitions,   etc.,   should   be   omitted    for   the 
present. 

753.  The  pupils  will  readily  see  that  the  words  "  Bought  of," 
used  in  Arts.  546  and  642,  are  inappropriate  in  bills  for  work 


74  MANUAL   FOE   TEACHERS 

done.  No.  5  may  be  made  out  in  the  form  here  shown  or  similar 
to  the  one  given  in  Art.  546.  See  Art.  642  for  a  bill  for  goods 
bought  at  different  times ;  or  use  the  heading  given  in  this  article. 

754.  What  has  been  said  about  percentage  in  Art.  746,  is  ap- 
plicable to  this  topic.  Such  children  as  hear  their  parents  talk 
of  savings-banks,  etc.,  know  sufficient  about  interest  for  the 
purposes  of  this  chapter.  No  rules  should  be  given. 

756.  The  pupils  should  deduce  their  own  rule  for  calculating 
the  area  of  a  right-angled  triangle. 

758.  In  Art.  653  the  pupils  have  been  taught  to  multiply  18f 
by  6  in  one  line ;  in  Art.  654,  they  have  learned  how  to  divide 
18J  by  2,  which  is  the  same  as  finding  -J-  of  18-|,  so  that  nothing 
new  is  here  presented. 

763-764.  Although  these  examples  are  not  strictly  practical, 
they  are  useful  in  giving  the  pupils  the  facility  necessary  to  per- 
form readily  operations  involving  fractions  or  decimals.  While 
it  is  not  necessary  to  work  them  all,  the  scholars  should  by  this 
time  have  acquired  such  expertness  in  the  fundamental  operations 
as  to  be  able  to  obtain  the  results  in  a  very  short  time. 

765.   See  Arithmetic,  Art.  591. 


XII 
NOTES  ON  CHAPTER  NINE 

The  technical  terms  used  in  denominate  number  work  should 
now  be  regularly  employed  by  teacher  and  pupil,  but  set  defi- 
nitions should  not  be  memorized.  The  scholars  should  be  re- 
quired to  arrange  their  work  properly,  and  to  perform  the 
various  operations  with  as  few  figures  as  are  consistent  with 
accuracy. 

767.  In  reducing  16  gal.  1  qt.  to  quarts,  the  pupil  should 
write  65  qt.  at  once.  He  multiplies  by  4,  saying  4  sixes  are  24, 
and  1  are  25  —  writing  the  5,  etc.  In  reducing  31^  gal.  to  quarts, 
the  work  should  occupy  but  a  single  line.  See  Arithmetic,  Art 
653. 

770.  No  special  rule  should  be  given  in  Nos.  33,  34,  and  35 
for  the  reduction  of  a  fractional  or  a  decimal  denominate  unit. 

773.  A  pupil  should  be  permitted  to  work  such  examples  as 
No.  2  in  his  own  way.  They  do  not  occur  frequently  enough  in 
practice  to  make  it  advisable  to  give  them  special  treatment  ; 
but  the  teacher  should  suggest,  as  in  other  exercises,  the  advis- 
ability of  shortening  the  work  by  indicating  operations  and  can- 
celling. Thus, 

12  min.  30sec.  =  12J  min.  =  ^  hr.  =  da.,  etc. 

9 
6'  60x24da* 


6.   750  Ib.  =  vWV  T.  =  f  T.  ;  $  5  x  5f  =  $  26.87^,  or  $  26.88. 

Ana. 
75 


76  MANUAL   FOR   TEACHERS 


7.  No.  of  tons  =  $18.76-^$5=  3.752;  .752  T.  =  (.  752  X  2000) 
lb.  =  1504  Ib.  Am.  3  T.  1504  Ib. 

8.  7  T.  296  Ib.  =  14296  Ib.  ;  ($  35.74  -*-  14296)  X  18748  =  Ans. 

9.  9  T.  1568  Ib.  =  19568  Ib.;   $48.92  -*-  19568  =  cost  per  Ib. 
$73.11  -H  ($48.92  -i-  19568),    or   ($  73.11  X  19568)  -t-  $48.92  = 
number  of  pounds.     Reduce  to  tons,  etc. 

774.  By  this  time,  the  pupils  should  know  how  to  add  com- 
pound numbers,  so  that  the  chief  duty  of  the  teacher  should  be 
to  see  that  the  operation  is  not  spun  out  too  much.     A  scholar 
of  this  grade  should  not  find  the  total  number  of  ounces  in  1  by 
adding  each  column  separately;  he  should  say  27,  36,  39  oz.,  or 
2  Ib.  7  oz.,  without  writing  anything  but  the  7  oz.,  which  is  put 
in  its  proper  column  and  2  Ib.  carried. 

In  4,  the  addition  of  the  units'  column  of  minutes  gives  a  sum 
of  15.  Since  minutes  are  changed  to  hours  by  dividing  by  60, 
which  ends  in  a  cipher,  the  units'  figure  of  the  remainder  will  be 
5,  so  that  this  figure  may  be  written  in  the  total.  Carrying  one, 
the  sum  of  the  tens'  column  is  11,  which  contains  6  once  with  a 
remainder  of  5.  This  is  written  in  its  place,  making  55  minutes, 
and  1  hour  is  carried.  The  two  columns  of  hours  are  added  in 
one  operation  —  21,  38,  43,  or  1  day  19  hours.  6  should  be 
treated  in  the  same  way,  no  side  work  being  permitted. 

In  7,  the  pounds  are  reduced  to  tons  by  dividing  by  2000,  so 
that  the  sum  of  the  units',  tens',  and  hundreds'  columns  of  pounds 
may  be  written  in  the  total,  the  sum  of  the  thousands'  column 
being  divided  by  2  to  reduce  to  tons. 

775.  Nothing  should  be  written  but  the  results.     In  27,  the 
addition  of  1  ton  to  1552  Ib.  will  change  only  one  figure  of  the 
latter,  and  this  change  can  be  carried  in  the  head.     In  29,  320 
rods  should  be  added  to  15  rods  mentally  and  24  rods  deducted 
from  the  sum,  only  the  answer  being  written. 

779.  In  dividing  5  bu.  by  4,  79,  the  answer  is  not  to  be  given 
as  1^  bu.  ;  the  division  should  be  continued  through  pecks.  The 
result  in  88  should  contain  weeks,  days,  hours,  and  minutes. 


NOTES  ON  CHAPTER   NINE  77 

784.  While  these  drills  seem  somewhat  difficult  for  mental 
work,  they  should  not  be  too  severe  for  children  that  have  been 
studying  arithmetic  for  over  five  years,  especially  if  the  previous 
drills  have  been  faithfully  attended  to.  The  ability  of  many 
children  to  handle  numbers  seems  to  decrease  after  the  fourth 
school  year,  the  greater  portion  of  the  subsequent  instruction 
being  given  to  new  topics  to  the  neglect  of  continued  practice  in 
the  fundamental  processes.  The  conscientious  teacher  should 
remember  that  the  bulk  of  the  mathematical  work  of  most  of  her 
scholars  after  they  leave  school  will  not  extend  much  beyond 
what  has  been  learned  in  the  first  four  years. 

The  ability  to  handle  at  sight  or  mentally  such  numbers  as 
are  here  given,  will  be  of  use  to  the  scholars  in  various  ways. 
The  average  pupil  attends  to  only  one  figure  at  a  time ;  and  he 
is  frequently  unable,  after  a  simple  addition  or  multiplication,  to 
see  that  his  answer  is  very  far  astray.  Practice  with  such  drills 
as  these,  and  in  the  sight  approximations,  will  enable  him  to 
test  his  work  in  such  a  way  as  to  detect  any  very  serious  error. 

Scholars  find  it  easier  to  add  or  subtract  such  numbers  as  163, 
8610,  etc.,  when  they  are  read  "one,  sixty-three  ;  "  "  eighty-six, 
ten ;  "  etc.  Following  the  order  in  which  the  figures  are  read 
seems  the  most  natural  way  in  mental  work.  When  a  pupil  is 
asked  to  find  the  sum  of  163  and  137,  he  is  less  likely  to  make 
mistakes  if  he  proceeds  in  this  way :  263,  293,  300 ;  adding  to 
the  first  number  — 163—100,  30,  and  7  in  the  order  in  which 
the  figures  are  repeated  to  him. 

786.  In  multiplying   21   by  15,  41  by  14,  etc.,  the   scholar 
generally  finds  it  easier  to  commence  with  the  tens :  15  twenties  are 
300,  15  ones  are  15—315 ;  14  forties  are  560,  and  14  are  574. 

48  X  16}  becomes  |  of  48  hundred;  32x37£  =  f  of  32  hun- 
dred, etc. 

787.  These  exercises  present  rather  more  difficulty,  and  are 
probably  not  so  useful,  qn  the  whole,  as  the  others.     For  this 
reason,  they  should  be  employed  as  sight  work  chiefly. 


78  MANUAL   FOE,   TEACHERS 

7BB.   In  13f  X5,  multiply  13  first  by  5,  and  then  f,  obtain- 
ing 65  +  3j,  or  68f.     In  dividing  24  by  2f,  reduce  both  to  thirds 
-  72  thirds  -**  8  thirds  =  72  -r-  8  =  9. 

790.  The  teacher  should  not  neglect  such  addition  exercises  as 
are  scattered  throughout  the  book. 

791.  It  happens  occasionally  in  multiplying  by 
a  mixed  number,  that  the  units'  figure  of  the  in- 
teger and  the  numerator  of  the  fraction  are  the 
same.     In  such  a  case,  a  few  figures  will  be  saved 
by  following  the  method  given  in  the  text-book, 
instead  of  writing  again  the  product  by  3  as  shown 

above.  etc. 

792.  The  product  by  100  may  be  placed  above  the  number,  if 
desired.     In  multiplying  by  1000,  the  multi- 
plicand is  subtracted  from  1000   times  itself.       97fi?I^X 

To  find  the  product  of  9832  by  990,  multiply  by       2761QQQ 

99,  and  annex  a  cipher  to  the  result.     Taking       2758239  Ans. 

one-fourth  of  268400  gives  the  answer  to  21. 

800.  The  pupils  should  find  for  themselves  in  5  the  number 
of  square  inches  in  a  square  foot,  etc.  A  drawing  is  asked  in 
the  first  part  of  14,  so  that  children  will  see  that  the  dimensions 
are  not  4x6.  The  short  method  of  finding  the  area  of  the  fence 
in  15,  by  multiplying  900  by  10,  should  not  be  given  yet:  the 
scholars  should  be  permitted,  for  the  present,  to  calculate  the  area 
of  one  part  at  a  time.  In  16,  it  is  suggested  that  the  area  of  the 
walk  be  ascertained  by  subtracting  from  the  whole  area  (250 
X  200)  sq.  ft.,  the  area  of  the  part  left  for  the  garden  (230  X  180) 
sq.  ft. ;  but  the  scholars  should  be  encouraged  to  calculate  the  sur- 
face of  the  walk  in  another  way,  such  as  by  taking  the  two  ends 
as  measuring  each  250  ft.  by  10  ft.,  and  the  sides  as  180  ft.  each 
by  10  ft.  The  number  of  square  feet  in  the  sidewalk  of  17  will 
be  (270  X  220)  -  (250  X  200)  ;  or  (270  X  10)  -f  (270  X  10)  +  (200 


NOTES   ON   CHAPTER   NINE  79 

X  10)  +  (200  X  10).     For  20,  a  modification  of  the  diagram  in 
Problem  20,  Art.  818,  is  desired. 

801.  To  show  pupils  what  is  required  in  21,  a  pasteboard  box, 
without  a  cover,  may  be  opened  out  as  is  represented  in  Problem 
2,  Art.  871,  the  upper  rectangle  (the  bottom  of  the  box)  repre- 
senting the  ceiling. 

802.  3.  The  sixth  dose  will  be  taken  at  7  o'clock,  the  second 
at  3  o'clock,  the  fourth  at  5  o'clock.     4.  He  works  6  days.    6.  A 
fence  6  ft.  long  will  require  2  posts ;  a  12-foot  fence  will  require 
3  posts ;  a  fence  120  ft.  long  will  require  21  posts. 

803.  In  finding  the  time  between  two  dates,  the  first  date  is 
excluded  except  when  the  contrary  is  expressly  specified. 

804.  11.    30  days +19  days.      12.   0  days  in  October +  30 
days  in  November  +  30  days  in  December. 

805.  1.    In  February,  there  are  (29-6)  days,  or  23  days. 
3  and  4.   Leap  year.     11.  Jan.  8,  15,  22,  29 ;  Feb.  5,  12, 19,  26 
are  Sundays.     The  man  works  30  days  in  January  and  28  in 
February,  less  8  Sundays  and  1  holiday. 

807^809.    See  Arts.  746  and  754. 

808.  3-13  should  be  worked  as  "sight"  exercises,  -J-  being 
used  for  25%,  ±  for  12}%,  etc. 

810.   First  compute  the  interest  for  one  year. 

1.  $3.60  for  a  year ;  |  of  $  3.60  for  2  months. 

2.  $  3.60  for  a  year ;  \  of  $  3.60  for  60  days. 

3.  $  5.00  for  a  year ;  $  5  X  2J-  for  2  yr.  6  mo. 

4.  $  6.00  for  a  year ;  -^  of  $  6  for  30  days. 

5.  $  9.00  for  a  year ;  J  of  $  9  for  90  days. 

81,2,  To  take  advantage  of  any  opportunities  for  cancellation 
that  may  be  offered,  this  method  is  given.  It  will  afterwards  be 
found  useful  in  calculating  the  principal,  the  rate,  or  the  time. 


80  MANUAL   FOE   TEACHERS 

Pupils  should  not  at  this  stage  be  taught  more  than  one 
method  of  finding  interest,  and  that  the  most  direct  and  the 
most  obvious  one. 

813.  2.  In  changing  2  mo.  12  da.  to  the  fraction  of  a  year,  it 
is  not  necessary  to  reduce  to  the  lowest  terms.  Change  the  time 
to  72  days,  and  write  360  underneath,  -gfo  ;  the  necessary  re- 
duction can  be  made  later  in  the  cancellation.  6.  Write  21 
months  as  ^  years,  canceling  afterwards. 

The  100  in  the  denominator  of  an  interest  example  should 
seldom  be  canceled,  except  as  a  whole  or  by  10. 

815.   1.    6  hr.  17  min.  5  sec.  =  22625  sec.;  3  hr.  15  min.  25 

sec.  =  11725  sec.  Ans.  ^Jff  =  $$£ 

2.    3mi.96rd.x3£  =  9mi.  288  rd.  -f  1  mi.  32  rd.  —  11  mi.  Ans. 

4.  A  furnished  ^  of  the  money,  and  should  receive  -J  of  $  1500, 
or  $750;  B  should  receive  |  of  $1500,  or  $500;  C  should  re- 
ceive £  of  $  1500. 

5.  If  5  T.  1000  Ib,  or  11000  lb.,  cost  $30.25,  1  Ib.  will  cost 
$30.25^-11000;  and  7  T.  320  lb.,  or  14320  lb,     <,__., 

will  cost  ($  30.25  +  11000)  X  14320.     Cancel. 


6.  25^x8ff-25^x8f.  Ans. 

7.  2  yr.  7  mo.  8  da.  —  31  mo.  8  da.  =  51-fomo.  =  31-^-  mo. 


10.  360  yd.@30^  cost  $108.  The  number  of  square  yards 
=  360  X  f£  =  360  X  f  =  270,  on  which  the  duty  at  8#  per  sq.  yd. 
will  be  8^  X  270,  or  $  21.60.  The  duty  on  the  value  will  be  50% 
of  $108,  or  |  of  $108,  or  $54;  the  total  duty  being  $21.60  -f- 
$54  -$75.60.  Ans. 

816.   3.  5  bbl.,  300  lb.  each,  @  5^  per  lb. 

4.  Interest  on  $  200  for  6  mo.  @  6%. 

5.  12  men  take  24  days  ;  how  long  will  24  men  take  ? 

6.  What  decimal  of  640  acres  is  320  acres  ? 

7.  20  thousand  bricks  @  $  20  per  M. 

8.  5600  lb.  @  87^  per  bu.  of  56  lb. 


NOTES  ON    CHAPTER    NINE  81 

9.    10  lb.  cost  $8  ;  find  cost  of  21  Ib. 
10.   Freight  on  20  hundred  lb.  @  70^  per  cwt. 

817.  2.    The  wall  8  yd.  X  4  yd.  contains  32  sq.  yd.  ;  the  door 
is  J  yd.  by  1±  yd.,  and  contains  4  sq.  yd.  ;  32  sq.  yd.  —  4  sq.  yd. 
=  28  sq.  yd.  Ans. 

3.  Number  of  square  inches  in  the  surface  of  the  widest  face 
=  8x4;  in  the  surface  of  one  side  =  8x2;  in  the  surface  of  end 
=  4x2. 

4.  (288  X  96)  H-  (8x4). 

5.  [(24  X  12)  X  (8x12)]  -+-[8x2]. 

6.  See  No.  20.     Make  four  rectangles  adjoining  each  other, 
each  8  inches  high  —  the  first  and  the  third  being  4  inches  wide  ; 
and  the  second  and  the  fourth,  2  inches  wide.     Above  and  below 
the  second,  and  connected  with  it,  draw  rectangles  2  inches  wide 
and  4  inches  high.     These  two  rectangles  may  be  drawn  above 
and  below  either  of  the  other  rectangles,  the  above  dimension 
being  used  if  drawn  above  and  below  the  fourth  ;  if  drawn  above 
and  below  the  first  or  the  third,  they  will  be  4  inches  wide  and 
2  inches  high.     The  pupils  should   be  permitted   to  make  the 
diagram  in  their  own  way,  and  they  should  be  encouraged  to 
make  one  that  differs  from  one  drawn  by  a  desk-mate. 

8.    The  number  of  rolls  will  be  (45x36)  -*-  (24x4).    Cancel. 

818.  The  scholars  should  make  this  table  without  any  assist- 
ance.    To  obtain  the  number  of  acres  in  a  square  mile,  indicate 
the  number  of  square  rods  in  a  square  mile,  320  x  320,  and 
divide  by  the  number  of  square  rods  in  an  acre,  160. 

14.  The  number  of  yards  =  (5  -f  3  +  4  +  7  -f  3  +  6  +  12  -f  10) 

X5*. 

15.  Original  dimensions  12  rods  X  13  rods,  making  area  156 
sq.  rd.     Present  area  =  156  sq.  rd.  —  (15  -f  21)  sq.  rd. 

19.   [|  of  (80 


819.   1.   43  yd.  =  (43  -*-  5£)  rd.  =  (43  -f-  Y)  rd.  =  (43  x 
rd.  =       rd 


82  MANUAL    FOR   TEACHERS 


2.  43yd.  =  7Isrrd.     ft  rd.  =  (  ^  X  5J)  yd.  =  (^  X  ^)  yd. 
=  4J  yd.  .Arcs.  7  rd.  4£  yd. 

3.  43  yd.  =  7  rd.  4J-  yd.  =  7  rd.  4  yd.  1£  ft. 

4.  43  yd.  =  7  rd.  4  yd.  1J  ft.  =  7  rd.  4  yd.  1  ft.  6  in. 

824.  34.    Carrying  1  to  the  column 

of  yards,  the  total  becomes  8  yd.  or  1  rd.  4  rd.  3    yd.  1  ft. 

2J-  yd.     Changing  J  yd.  to  1  ft.  6  in.,  and  9  rd.  4    yd.  2  ft. 

adding  this  to  17  rd.  2  yd.  1  ft.  6  in.,  3  rd.               1  ft.  6  in. 

the  accompanying  answer  is  obtained.  17  r(}.  2J  yd.  1  ft.  6  in. 

38.    8  rd.  Oyd.  1ft.—  2  rd.  Oyd.  2ft.  17  rd.  2    yd.  1  ft.  6  in. 

=  5  rd.  41  yd.  2  ft.  =  5  rd.  4  yd.  2  ft.  +1.  yd.  =  1  ft.  6  in. 

+  1  ft.  6  in.  -  5  rd.  5  yd.  6  in.  Am.  17  rd.  3    yd.  Am. 

40.  5  rd.   4  yd.   2   ft.  X  4  =  23  rd. 

1J  yd.  2  ft.  =  23  rd.  1  yd.  2  ft.  +  1  ft.  6  in.  =  23  rd.  2  yd.  6  in. 

825.  10.    The  other  dimensions  would  be  8  ft.  and  4  ft.,  or 
16  ft.  and  2  ft. 

14.  Number  of  cubic  yards  =  ^8-  X  &£•  X  f.     Cancel. 

15.  i  (yd.)  X  2  (yd.)  X  width  (yd.)  =  1  (cu.  yd.)  ;    or  £  X  2 
Xa?=l;  a;  =  1  ;  lyd.  ,4ws. 

16.  A  gallon    contains   231  cu.  in.  ;    a   cubic   foot   contains 
1728  cu.  in.     1  cu.  ft.  =  (1728  -5-  231)  gal.  =  7£ft  gal.  -  about 
7-J-  gal.  Am. 

17.  About  1J-  cu.  ft.  Am. 

18.  Number  of  gallons  (21  X  15  X  22)  H-  231. 

19.  The  decimal  in  the  denominator  is  re-      36  x  28  X  6400 
moved  one  place  to  the  right,  and  a  cipher  is  2150  0  4 
annexed  to  64  in  the  numerator. 

NOTE.  —  2150.4  cu.  in.  is  used  instead  of  2150.42  cu.  in.,  the  more  correct 
equivalent,  because  the  former  is  divisible  by  6,  7,  8,  etc. 

25.    [$  6.40  X  (40x161)  X4x3]^-24f.     Cancel. 


NOTES  ON  CHAPTER  NINE  83 

826.  3.  At  7£  gal.  to  cu.  ft.,  a  tank  of  150  gal.  will  contain 
(150  H-  7£)  cu.  ft.  =  20  cu.  ft.  The  dimensions  will  be  2  ft.  X  2  ft. 
X  5  ft.,  or  4  ft.  X  1  ft.  X  5  ft.,  etc.,  etc. 

4.  At  1J  cu.  ft.  to  a  bushel,  the  bin  will  contain  1J  cu.  ft. 
X  100  =  125  cu.  ft.     The  dimensions  will  be  5  ft.  X  5  ft.  X  5  ft., 
or  5  ft.  by  10  ft.  by  2J  ft.,  etc.,  etc. 

5.  1000  bricks  will  build  (1000  -4-  20)  cu.  ft.     A  wall  1  ft. 
thick  can  be  10  ft.  long  and  2  ft.  high,  or  5  ft.  long  and  4  ft. 
high,  etc. 

6.  10  yd.  X  5  yd.  X  2  yd.  (30  ft.  X  15  ft.  X  6  ft.),  4  yd.  X 
5  yd.  X  5  yd.,  etc. 

7.  A  gallon  weighs  about  8  Ib. ;  a  pint  about  1  Ib. 

8.  A  cubic  foot  of  iron  weighs  about  7  times  64  pounds. 

9.  About  %  of  $800. 

10.    About  4  years'  interest. 

831.  5.   See  Arithmetic,  Art.  642. 

6.  See  Arts.  829-830. 

7.  Three  inches  square  =  (3x3)  sq.  in. 

832.  1.  See  Art.  1022,  No.  15. 

4.    Including  Sept.  19,  the  time  is  (28  +  30  +  31  +  30  -f  31 

+  31 +  19)  days. 

0 

833.  6.   The  written  analysis  of  an  arithmetic  example  should 
be  required  occasionally  as  an  exercise  in  composition. 

7.  7000  gr.  X  2J  =  number  of  grains  in  2  Ib.  14  oz.  Dividing 
by  480  grains,  the  number  of  Troy  ounces  is  obtained  —  (7000  x 
2J)-4-480.  Multiplying  $1.80  by  this  number,  the  cost  of  the 
urn  is  ascertained  —  ($  1.80  X  7000  X  2J)  -*•  480. 

835.  8.  Since  the  denominator  of  a  fraction  indicates  the 
number  of  parts  into  which  a  thing  is  divided,  a  larger  denomina- 


84  MANUAL   FOR   TEACHERS 

tor  indicates  a  greater  number  of  parts,  and,  therefore,  smaller 


ones. 


839.   1.    If  three-fifths  of  a  bbl.  cost  $2.13,  six-fifths  will  cost 
twice  $2.  13. 

8.    Commission  at  1%  would  amount  to  $3;  at  \°fo,  it  is 
$1.50. 

18.    f=18^;  i  =  6^;  i  or  J  of  J,  =  -J-  of  6£ 


841.   Add   without   re-writing   the   fractions    reduced    to    a 
common  denominator. 


845.  10.  75%  of  A,  or  f  of  -ft.  or  ^,  is  sold  for  $1710. 
Factory  is  worth  $  1710-5-^. 

13.  First  piece  contains  (20  X  f)  sq.  yd.,  or  15  sq.  yd.  Width 
of  second  piece  in  yards  =  15  -*-  12. 

16.  8  men  and  5  boys  =  8  men  +  2|-  men  =  10-J-  men.     If  7 
men  do  a  piece  of  work  in  1GJ-  days,  1  man  will  do  it  in  10J-  days 
X  7,  and  10£  men  will  do  it  in  (10J  days  X  7)  -5-  10J.     Cancel. 

17.  Each  of  the  six  square  faces  of  a  cube  contains  (6  X  6)  sq.  in., 
or  36  sq.  in.  ;  the  whole  surface  will  be,  therefore,  36  sq.  in.  X  6. 
Each  face  contains  (-J-  X  £)  sq.  ft.  =  \  sq.  ft.  ;  whole  surface  = 
\  sq.  ft.  X  6. 

Contents  in  cu.  in.  =  6  X  6  X  6  ;  in  cu.  ft.  =  \  X  |  X  \. 

18.  f-J-  in  water  and  \\  in  mud  =  |-£,  leaving  -fa  above  water, 
or  5  ft.     Length  of  post  =  5  ft.  -5-  -fa. 

19.  [(10  X  9)  -f  (12  X  10)  +  (8  X  11)  +  (6  x  12)  -f  (2  X  13)  + 
(1X14)]+-  [10  +  12  +  8  +  6  +  2+1]. 

21.  From  Oct.  25  to  Dec.  31,  inclusive,  there  are  7  +  30  +  31, 
or  68  days  ;  Oct.  30  is  Sunday  ;  also  Nov.  6,  13,  20,  '27  ;  Dec.  4, 
11,  18,  and  25  —  9  Sundays,  Election  Day,  and  Thanksgiving  to 
be  deducted,  or  11  days,  leaving  57  days,  at  $3|-  per  day. 

23.    12  Ib.  tea  cost  $2.80  +  $2.00,  or  $4.80;  value  per  Ib. 


NOTES  ON   CHAPTER  NINE  85 

24.  House  and  lot,  or  3|  lots  +  1  lot,  or  4|  lots  =  $8100;  1 
lot  =  $8100-*-4i  =  $1800;  house =$  1800  x  3f 

$18  x  (20  x  12)  x  (15  x  12)  x  (6  x  12) 
1000  x  8  x  4  x  2 

27.  36   yd.   8  in.  =  1304  in.;  13   yd.  1  ft.  9  in.  =489   in.; 
quantity  left  =  1304  in.  —  489  in.  =  815  in. ;  fraction  left  =  &£? 
=  | ;  decimal  left  =  .625 ;  per  cent  left  =  62f  Ans. 

28.  Assessed  value  =  80%  of  $  30000  =  $  24000.     Taxes  on 
24  thousand  dollars  =  $21. 60  X  24. 

846.  2.  See  Arithmetic,  Art.  1251 ;  angles  E,  F,  G,  and  H, 
and  M,  N,  0,  and  P. 

3.  Art.  1251 ;  angles  A  and  B,  C  and  D. 

4.  Angles  /  and  J,  K  and  L.     The  scholars  should  under- 
stand that  two  lines  can  be  perpendicular  without  one  being  a 
horizontal  line  and  the  other  a  vertical  line. 

5.  The  size  of  an  angle  does  not  depend  upon  the  length  of 
the  lines  that  form  the  angle.     Two  short  lines  may  meet  at  a 
very  obtuse  angle,  and  two   long  lines  may  form  a  very  acute 
angle. 

13.  If  the  pupils  have  in  their  drawing  lessons  constructed 
triangles  by  means  of  compasses,  these  may  be  used ;  otherwise, 
let  them  manage  as  best  they  can,  no  great  accuracy  being 
required. 

15.  Children  are  accustomed  to  seeing  an  isosceles  triangle  in 
only  one  position :  they  should  learn  that  if  a  triangle  has  two 
equal  sides,  it  is  isosceles,  no  matter  whether  the  unequal  side  is 
vertical,  horizontal,  or  oblique. 

16-22.  Accustom  the  scholars  to  the  occasional  employment 
of  an  oblique  line  as  a  base  in  constructing  squares,  rectangles, 
etc.  See  Arithmetic,  Art.  1265.  A  card  may  be  used  to  make  a 
square  corner. 


86  MANUAL   FOR   TEACHERS 

24.  See  Arithmetic,  Art.  929,  No.  8,  for  a  rectangle,  a  rhombus, 
and  a  rhomboid,  having  equal  bases  and  equal  altitudes.     No.  5 
shows  three  rhomboids  of  equal  bases  and  equal  altitudes,  but 
differing  in  shape. 

25.  See  Art.  929,  No.  8. 

847.   1.    (1  of  15  X  20)  sq.  in.     The  length  of  the  third  side 
does  not  enter  into  the  computation. 

6.  Let  the  scholars  find  the  area  of  the  rectangle,  66  ft.  by 
63  ft.,  and  the  two  triangles,  31  ft.  each  by  63  ft.,  and  find  the 
sum  of  the  areas.    Then  lead  them  to  see  that  bringing  the  right- 
hand  triangle  to  the  left  of  the  rhombus  would  make  a  rectangle 
97  ft.  by  63  ft.,  whose  area  is  the  sum  above  found. 

7.  Find  the  area  in  square  meters,  saying  nothing  more  about 
the  meter  than  that  it  is  largely  used  on  the  continent  of  Europe, 
and  is  a  little  longer  than  a  yard. 

8-10.  Give  no  rules  yet  for  calculating  the  areas  of  trapezoids 
and  trapeziums.  Let  the  pupils  ascertain  the  areas  of  the  figures 
from  the  data  supplied. 


XIII 
NOTES  ON  CHAPTER  TEN 

The  formal  study  of  algebra  belongs  to  the  high-school ;  but 
some  so-called  arithmetical  problems  are  so  much  simplified  by 
the  use  of  the  equation  that  it  is  a  mistake  for  a  teacher  not  to 
avail  herself  of  this  means  of  lightening  her  pupils'  burdens. 

In  beginning  this  part  of  her  mathematical  instruction,  the 
teacher  should  not  bewilder  her  scholars  with  definitions.  The 
necessary  terms  should  be  employed  as  occasion  requires,  and 
without  any  explanation  beyond  that  which  is  absolutely  neces- 
sary. 

849.  Very  young  pupils  can  give  answers  to  most  of  these 
questions ;  so  that  there  will  be  no  need,  for  the  present,  at  least, 
of  introducing  a   number  of  axioms   to  enable   the   scholar   to 
obtain  a  result  that  he  can  reach  without  them. 

850.  Pupils  will  learn  how  to  work  these  problems  by  work- 
ing a  number  of  them.     They  may  need  to  be  told  that  x  stands 
for  la;;  and  that,  as  a  rule,  only  abstract  numbers  are  used  in 
the  equations,  the  denomination  —  dollars,  marbles,  etc.  —  being 
supplied  afterwards. 

While  the  scholars  should  be  required  to  furnish  rather  full 
solutions  of  the  earlier  problems,  they  should  be  permitted  to 
shorten  the  work  by  degrees,  writing  only  whatever  may  be 
necessary. 

4.  *-f22:  =  54.  8.   x  +  2x  +  6*  =  27000. 

5.  a;  +  5x=78.  9. 

6.  lx  +  5.r=156.  10.   x 

7.  9*-32:  =  66.  11.   * 

87 


88  MANUAL    FOR   TEACHERS 


12. 

13.    Let  x  =  the  fourth  ;  then  4  x  =  the  third,  12  x  =  the  second, 
and  24  a;  =  the  first. 


14.  a?  =the  second,  2  a;  =  first,  9  x  —  third. 

15.  5a?  +  4a?  =  81.  17.    4a;  = 

16.  24a;  =  456.  19.    3  a;  +  4  a;  =  175. 
20.    Let  x  =  each  boy's  share  ;  2  x  —  each  girl's  share. 


21.   a;  —  number   of  days    son  worked;  2  x  =  number   father 
worked.     3  x  =  son's  earnings  ;  8  x  =  father's  earnings. 


22.    x  =  number   of  dimes;    2  x  =  number   of  nickels;    Qx  = 
number  of  cents. 


or 

23.  15a;  —  12z  = 
24. 

25.  Let  x  =  cost  of  speller  ;  then  3  x  —  cost  of  reader. 

26.  Let  x  =  smaller  ;  then  5  x  =  larger. 

27.  Let  x  =  Susan's  number  ;  2  x  =  Mary's  ;  3  x  =  Jane's. 

851.  10  :  %x  is  the  same  as  |- 

o 

852.  Pupils  already  know  that  J  means  3-?-  4,  so  that  they 
can  understand  that  —  means  3  x  -*-  4,  or  \  of  3  x.     When  %  of 
something  (3  x)  is  24,  the  whole  thing  (3  a?)  must  be  4  times  24, 

or  96  ;  that  is,  when  —  =  24,  3  x  =  96. 

4 

When  ^  =  24,  2^  =  24x3,  or  72. 

3 

When  i2  =  20,  4  z  =  20  X  5,  or  100. 

o 

From  these  examples  can  be  formulated  the  rule  for  disposing 
of  a  fraction  in  one  term  of  an  equation,  which  is,  to  multiply 


NOTES  ON   CHAPTER  TEN  89 

both  terms  by  the  denominator  of  the  fraction.  In  changing  the 
first  term  of  the  equation,  —  ^  =  24,  to  3#,  it  has  been  multi- 
plied by  4,  so  that  the  second  term  must  also  be  multiplied  by  4. 

853.  In  solving  these  examples  by  the  algebraic  method  of 
"  clearing  of  fractions,"  attention  may  be  called  to  its  similarity 
to  the  arithmetical  method.  To  find  the  value  of  y  in  2,  the 
pupil  multiplies  8  by  5  and  divides  the  product  by  2  ;  as  an  ex- 
ample in  arithmetic,  he  would  divide  8  by  -J,  that  is,  he  would 
multiply  8  by  f;  the  only  difference  being  that  by  the  latter 
method  he  would  cancel. 

While  —  *  =  8  may  be  changed  to  *-  =  4  by  dividing  both  terms 
o  o 

by  2,  beginners  are  usually  advised  to  begin  by  "  clearing  of 
fractions,"  short  methods  being  deferred  to  a  later  stage. 


854. 


6  may  be  written       -f       =  92. 

U  I 


8.    2Jz  should  be  reduced  to  an  improper  fraction,  making  the 

equation,  —  -  =  115.     Make  similar  changes  in  12,  14,  18,  and  20. 

8 

855.  2.   *  +  ^ 

2 

6.    |  +  |  =  ?|? 
6.    §£_?*=  15. 


9.  Let  5 a;  =  numerator  ;  1x  =  denominator.  7ar  —  5ar  =  24; 
2x  =  24 ;  x  =  12.  The  numerator,  5x,  will  be  5  times  12,  or  60 ; 
the  denominator  will  be  84 ;  and  the  fraction,  £J.  Ans. 

10.   Let  x  =  greater ;      =  less. 


90  MANUAL   FOR  TEACHERS 

Clearing  of  fractions,         7x  +  x  =  3360, 

Sx  =  3360, 
x  =  420,  the  greater  number, 

£  -  60,  the  less. 

Or,  let  x  =  less  ;  7  x  =  greater. 

x  +*lx  =  480, 
8  #  =  480, 

x  =  60,  the  less, 
7  x  -=420,  the  greater. 

The  employment  of  the  latter  plan  does  away  with  fractions 
in  the  original  equation. 

11.    30z-  x  =  522,  or  x  -—-522. 

30 

13.    Let  x  —  number   of    plums;    4x  =  number    of  peaches. 
Then  2x  will  be  cost  of  plums,  and  12  a;  the  cost  of  the  peaches. 

2x+l2x  -70. 
15.  a?-y  =  80. 

•:f:l-*    ;  •  *':i» 

18.  x  +  l±x  +  (l±x  X  3|)  -  15. 

ar  +  —  +  5o?=15. 

2 

19.  Let  a?  =  price  per  yard  of  the  48-yard  piece  ;  2  a;  =  price 
per  yard  of  the  36-yard  piece  ;  48  x  will  be  the  total  cost  of  one, 
and  72  x,  of  the  other. 

72^  =  240. 


20.    160  x  +120^  =  840. 

856.    The  pupils  should  be  permitted  to  give  these  answers 
without  assistance. 

In  Art.  857  is  explained  what  is  meant  by  "  transposing." 


NOTES   ON   CHAPTER  TEN  91 

858.  While  these  exercises  are  so  simple  that  they  can  be 
worked  without  a  pencil,  they  should  be  used  to  show  the  steps 
generally  taken  in  more  complicated  equations.  _ 

In  1,  for  instance,  the  work  should  take  the 
form  here  indicated,  only  a  single  step  being  _ 

taken  at  a  time.     In  19,  the  first  step  is  to 
clear  the  equation  of  fractions  by  multiplying  by  6  ;  the  second 

step   is  to   transpose   the    unknown 

2x  —   6  =  16  +  |  —  I        quantities  to   the   left  side    of   the 
•i  2    —36  =  96  +  3    —  *>      e(luation'  ancl  tlie  known  quantities 

12*  -  2*  +  2*  =  96  +  36   *  the  "ght  ;  ,  *«  *ird  steP.ia  to  "T" 
11    —  132  unknown    quantities    into 

—  12  °ne>  an(*  *°  ma^e  a  8im^ar  combina- 

tion  of   the   known   quantities;  the 
last  step  is  to  find  the  value  of  x. 

After  a  little  more  familiarity  with  exercises  of  this  kind,  the 
pupil  can  take  short  cuts  with  less  danger  of  mistakes  ;  for  the 
present,  however,  it  will  be  safer  to  proceed  in  the  slower  way. 


859.  5.   *  +  (*+  75)  +  *  +  (*+  75)  =  250. 

a:  +  *  +  #  +  *  =  250  -75  -75. 


NOTE.  —  The  parentheses  used  here  are  unnecessary.     They  are  employed 
merely  to  show  that  x  +  75  is  one  side  of  the  field. 


6.   x  +  (x+8)  =  86.  9.   x  +  x+  72  =  96. 

X  _X 

3      4 


7.   x  +  x + 318  =  2436.  10.   *----=45. 


8.  z+  £+7  =  100. 

a 

II.   x  —  one  part  ;  2x  —  6  =  other  part. 


12.   x  =  John's  money  ;  x  -f  5  =  William's  money. 
3*  +  15  +  5*  =103. 


92  MANUAL    FOE   TEACHERS 

13.  Let  x  =  price  of  a  horse  ;         x  —  80  =  price  of  a  cow  ; 

4  x  =  cost  of  four  horses  ;    3  x  —  240  =  cost  of  three  cows. 
4#  +  3#    -240  —  635, 
7  x  =  635  +  240  =  875, 

x  =  125,  price,  in  dollars,  of  a  horse  ; 
x  —  80  =    45,  price,  in  dollars,  of  a  cow. 
Other  pupils  may  solve  the  problems  in  this  way  : 

x  =  price  of  a  cow  ;  x  +  80  =  price  of  a  horse. 
3x+  4^  +  320  =  635, 
7  x  =  635  -320  =315, 

x  =    45,  price,  in  dollars,  of  a  cow  ; 
x  +  80  =  125,  price,  in  dollars,  of  a  horse. 

14.  x  =  number   of    dimes;    x  +  11  =  number   of    five-cent 
pieces;    10  x  =  value  of   dimes  (in  cents);    5  x  +  55  =  value  of 
five-cent  pieces. 


15.    x  =  greater  ;  x  —  48  =  less. 

a;  -j-*-  48  =  100. 
Or,  x  =  less  ;  x  +  48  =  greater. 


17.  x  —  share  of  the  first  ; 

x  +  2400  =  share  of  the  second  ; 

x  +  2400  +  2400  =  share  of  the  third. 

x  +  x  +  2400  +  x  +  2400  +  2400  =  18000. 

18.  Let  x  =  less  ;  x  +  33  =  greater. 


Bringing  known  quantities  to  the  left  side  of  the  equation,  and 
the  unknown  quantities  to  the  right, 

33-ll  =  3ar-3r, 
22  =  2a?, 


NOTES  ON  CHAPTER  TEN  93 

Or,  *-3*=ll-33, 

—  2x  =  —  22. 
Changing  signs  of  both  terms, 

2*  =22, 
*  =11. 

This  problem  may  also  be  worked  in  this  way  : 
x  =  less ;  3x  -f- 11  =  greater. 
3*  4- 11 -a;  =33. 

19.  x  =  number  of  5-cent  stamps  ;  x  -f- 15  =  number  of  2-cent 
stamps  ;  x  -f  30  =  number  of  postal  cards. 

5a;-f2a;-f30-f2f  +  30  =  100. 

20.  x  =  number  of  horses ;  x  + 17  =  number  of  cows ;  2x  -f-  39 
=  number  of  sheep. 

x  +  x  -f  1 7  +  2  x  +  39  =  88 . 


SUPPLEMENT 


DEFINITIONS,   PRINCIPLES,   AND   RULES 

A  Unit  is  a  single  thing. 

A  Number  is  a  unit  or  a  collection  of  units. 

The  Unit  of  a  Number  is  one  of  that  number. 

Like  Numbers  are  those  that  express  units  of  the  same  kind. 

Unlike  Numbers  are  those  that  express  units  of  different  kinds. 

A  Concrete  Number  is  one  in  which  the  unit  is  named. 

An  Abstract  Number  is  one  in  which  the  unit  is  not  named. 

Notation  is  expressing  numbers  by  characters. 

Arabic  Notation  js  expressing  numbers  by  figures. 

Roman  Notation  is  expressing  numbers  by  letters. 

Numeration  is  reading  numbers  expressed  by  characters. 

The  Place  of  a  figure  is  its  position  in  a  number. 

A  figure  standing  alone,  or  in  the  first  place  at  the  right  of  other 
figures,  expresses  ones,  or  units  of  the  first  order. 

A  figure  in  the  second  place  expresses  tens,  or  units  of  the 
second  order. 

A  figure  in  the  third  place  expresses  hundreds,  or  units  of  the 
third  order ;  and  so  on. 

A  Period  is  a  group  of  three  orders  of  units,  counting  from  right 
to  left. 

RULE  FOR  NOTATION.  —  Begin  at  the  left,  and  write  the  hun- 
dreds, tens,  and  units  of  each  period  in  succession,  filling  vacant 
places  and  periods  with  ciphers. 


ii  SUPPLEMENT 

RULE  FOB,  NUMERATION.  —  Beginning  at  the  right,  separate  the 
number  into  periods. 

Beginning  at  the  left,  read  the  numbers  in  each  period,  giving 
the  name  of  each  period  except  the  last. 

ADDITION 

Addition  is  finding  a  number  equal  to  two  or  more  given  num- 
bers. 

Addends  are  the  numbers  added. 

The  Sum,  or  Amount,  is  the  number  obtained  by  addition. 

PRINCIPLE.  —  Only  like  numbers,  and  units  of  the  same  order 
can  be  added. 

RULE.  —  Write  the  numbers  so  that  units  of  the  same  order  shall 
be  in  the  same  column. 

Beginning  at  the  right,  add  each  column  separately,  and  write 
the  sum,  if  less  than  ten,  under  the  column  added. 

When  the  sum  of  any  column  exceeds  nine,  write  the  units  only, 
and  add  the  ten  or  tens  to  the  next  column. 

Write  the  entire  sum  of  the  last  column. 

SUBTRACTION 

Subtraction  is  finding  the  difference  between  two  numbers. 
The  Subtrahend  is  the  number  subtracted. 
The  Minuend  is  the  number  from  which  the  subtrahend  is  taken. 
The  Bemainder,  or  Difference,  is  the  number  left  after  subtracting 
one  number  from  another. 

PRINCIPLES.  —  Only  like  numbers  and  units  of  the  same  order 
can  be  subtracted. 

The  sum  of  the  difference  and  the  subtrahend  must  equal  the 
minuend. 

RULES.  —  I.  Write  the  subtrahend  under  the  minuend,  placing 
units  of  the  same  order  in  the  same  column. 


DEFINITIONS,    PRINCIPLE>T£TTOLRflLJ5a^^  111 


Beginning  at  the  right,  find  the  number  that  must  be  added  to 
the  first  figure  of  the  subtrahend  to  produce  the  figure  in  the  corre- 
sponding order  of  the  minuend,  and  write  it  below.  Proceed  in 
this  way  until  the  difference  is  found. 

If  any  figure  in  the  subtrahend  is  greater  than  Oie  corresponding 
figure  in  the  minuend,  find  the  number  that  must  be  added  to  the 
former  to  produce  the  latter  increased  by  ten ;  then  add  one  to  the 
next  order  of  the  subtrahend  and  proceed  as  before. 

II.  Beginning  at  the  units'  column,  subtract  each  figure  of  the 
subtrahend  from  the  corresponding  figure  of  the  minuend  and 
write  the  remainder  below. 

If  any  figure  of  the  subtrahend  is  greater  than  the  corresponding 
figure  in  the  minuend,  add  ten  to  the  latter  and  subtract ;  then, 
(a)  add  one  to  the  next  order  of  the  subtrahend  and  proceed  as 
before ;  or,  (b)  subtract  one  from  the  next  order  of  the  minuend 
and  proceed  as  before. 

MULTIPLICATION 

Multiplication  is  taking  one  number  as  many  times  as  there  are 
units  in  another  number. 

The  Multiplicand  is  the  number  taken  or  multiplied. 

The  Multiplier  is  the  number  that  shows  how  many  times  the 
multiplicand  is  taken. 

The  Product  is  the  result  obtained  by  multiplication. 

PRINCIPLES.  —  The  multiplier  must  be  an  abstract  number. 
The  multiplicand  and  the  product  are  like  numbers. 
The  product  is  the  same  in  whatever  order  the  numbers  are 
multiplied. 

RULE.  —  Write  the  multiplier  under  the  multiplicand,  placing 
units  of  the  same  order  in  the  same  column. 

Beginning  at  the  right,  multiply  the  multiplicand  by  the  number 
of  units  in  each  order  of  the  multiplier  in  succession.  Write  the 


IV  SUPPLEMENT 

figure  of  the  lowest  order  in  each  partial  product  under  the  figure 
of  the  multiplier  that  produces  it.     Add  the  partial  products. 

To  multiply  by  10,  100,  1000,  etc. 

KULE.  —  Annex  as  many  ciphers  to  the  multiplicand  as  there 
are  ciphers  in  the  multiplier. 

DIVISION 

Division  is  finding  how  many  times  one  number  is  contained  in 
another,  or  finding  one  of  the  equal  parts  of  a  number. 
The  Dividend  is  the  number  divided. 
The  Divisor  is  the  number  contained  in  the  dividend. 
The  Quotient  is  the  result  obtained  by  division. 

PEJNCIPLES.  —  When  the  divisor  and  the  dividend  are  like  num- 
bers, the  quotient  is  an  abstract  number. 

When  the  divisor  is  an  abstract  number,  the  dividend  and  the 
quotient  are  like  numbers. 

The  product  of  the  divisor  and  the  quotient,  plus  the  remainder, 
if  any,  is  equal  to  the  dividend. 

RULE.  —  Write  the  divisor  at  the  left  of  the  dividend  with  a  line 
between  them. 

Find  how  many  times  the  divisor  is  contained  in  the  fewest  fig- 
ures on  the  left  of  the  dividend,  and  write  the  result  over  the  last 
figure  of  the  partial  dividend.  Multiply  the  divisor  by  this  quotient 
figure,  and  write  the  product  under  the  figures  divided.  Subtract 
the  product  from  the  partial  dividend  used,  and  to  the  remainder 
annex  the  next  figure  of  the  dividend  for  a  new  dividend. 

Divide  as  before  until  all  the  figures  of  the  dividend  have  been 
used. 

If  any  partial  dividend  will  not  contain  the  divisor,  write  a 
cipher  in  the  quotient,  and  annex  the  next  figure  of  the  dividend. 

If  there  is  a  remainder  after  the  last  division,  write  it  after  the 
quotient  with  the  divisor  underneath. 


DEFINITIONS,    PRINCIPLES,   AND   RULES  V 

FACTORING 

An  Exact  Divisor  of  a  number  is  a  number  that  will  divide  it 
without  a  remainder. 

An  Odd  Number  is  one  that  cannot  be  exactly  divided  by  two. 

An  Even  Number  is  one  that  can  be  exactly  divided  by  two. 

The  Factors  of  a  number  are  the  numbers  that  multiplied  to- 
gether produce  that  number. 

A  Prime  Number  is  a  number  that  has  no  factors. 

A  Composite  Number  is  a  number  that  has  factors. 

A  Prime  Factor  is  a  prime  number  used  as  a  factor. 

A  Composite  Factor  is  a  composite  number  used  as  a  factor. 

Factoring  is  separating  a  number  into  its  factors. 

To  find  the  Prime  Factors  of  a  Number. 

RULE.  —  Divide  the  number  by  any  prime  factor.  Divide  the 
quotient,  if  composite,  in  like  manner;  and  so  continue  until  a 
prime  quotient  is  found.  The  several  divisors  and  the  last  quotient 
will  be  the  prime  factors. 

CANCELLATION 

Cancellation  is  rejecting  equal  factors  from  dividend  and  divisor. 
PRINCIPLE.  —  Dividing  dividend  and  divisor  by  the  same  num- 
ber does  not  affect  the  quotient. 

GREATEST  COMMON  DIVISOR 

A  Common  Factor  (divisor  or  measure)  is  a  number  that  is  a 
factor  of  each  of  two  or  more  numbers. 

A  Common  Prime  Factor  is  a  prime  number  that  is  a  factor  of 
each  of  two  or  more  numbers. 

The  Greatest  Common  Factor  (divisor  or  measure)  is  the  largest 
number  that  is  a  factor  of  each  of  two  or  more  numbers. 

Numbers  are  prime  to  each  other  when  they  have  no  common 
factor. 


VI  SUPPLEMENT 

The  greatest  common  divisor  of  two  or  more  numbers  is  the 
product  of  their  common  prime  factors. 

PEINCIPLES.  —  A  common  divisor  of  two  numbers  is  a  divisor 
of  their  sum,  and  also  of  their  difference. 

A  divisor  of  a  number  is  a  divisor  of  every  multiple  of  that 
number ;  and  a  common  divisor  of  two  or  more  numbers  is  a 
divisor  of  any  of  their  multiples. 

To  find  the  Common  Prime  Factors  of  Two  or  More  Numbers. 

RULE.  —  Divide  the  numbers  by  any  common  prime  factors, 
and  the  quotients  in  like  manner,  until  they  have  no  common 
factor ;  the  several  divisors  are  the  common  prime  factors. 

To  find  the  Greatest  Common  Divisor  of  Numbers  that  are  Easily 
Factored. 

RULE.  —  Separate  the  numbers  into  their  prime  factors ;  the 
product  of  those  that  are  common  is  the  greatest  common  divisor. 

To  find  the  Greatest  Common  Divisor  of  Numbers  that  are  not 
Easily  factored. 

RULE.  —  Divide  the  greater  number  by  the  less;  then  divide 
the  last  divisor  by  the  last  remainder,  continuing  until  there  is  no 
remainder.  The  last  divisor  is  the  greatest  common  divisor. 

If  there  are  more  than  two  numbers,  find  the  greatest  common 
divisor  of  two  of  them;  then  of  that  divisor  and  another  of  the 
numbers  until  all  of  the  numbers  have  been  used.  The  last  divisor 
is  the  greatest  common  divisor. 

LEAST  COMMON  MULTIPLE 

A  Multiple  of  a  number  is  a  number  that  exactly  contains  that 
number. 

A  Common  Multiple  of  two  or  more  numbers  is  a  number  that 
is  a  multiple  of  each  of  them. 

The  Least  Common  Multiple  of  two  or  more  numbers  is  the 
smallest  number  that  is  a  common  multiple  of  them. 


DEFINITIONS,    PRINCIPLES,   AND   RULES  vii 

PRINCIPLES.  —  A  multiple  of  a  number  contains  all  the  prime 
factors  of  that  number. 

A  common  multiple  of  two  or  more  numbers  contains  each  of 
the  prime  factors  of  those  numbers. 

The  Least  Common  Multiple  of  two  or  more  numbers  contains 
only  the  prime  factors  of  each  of  the  numbers. 

To  find  the  Least  Common  Multiple  of  Two  or  More  Numbers. 

RULE.  —  Divide  by  any  prime  number  that  is  an  exact  divisor  of 
two  or  more  of  the  numbers,  and  write  the  quotients  and  undivided 
numbers  below.  Divide  these  numbers  in  like  manner,  continuing 
until  no  two  of  the  remaining  numbers  have  a  common  factor. 
The  product  of  the  divisors  and  remaining  numbers  is  the  least 
common  multiple. 

FRACTIONS 

A  Fraction  is  one  or  more  of  the  equal  parts  of  anything. 

The  Unit  of  a  Fraction  is  the  number  or  thing  that  is  divided 
into  equal  parts. 

A  Fractional  Unit  is  one  of  the  equal  parts  into  which  the  num- 
ber or  thing  is  divided. 

The  Terms  of  a  Fraction  are  its  numerator  and  its  denominator. 

The  Denominator  of  a  fraction  shows  into  how  many  parts  the 
unit  is  divided. 

The  Numerator  of  a  fraction  shows  how  many  of  the  parts  are 
taken. 

A  fraction  indicates  division ;  the  numerator  being  the  divi- 
dend and  the  denominator  the  divisor. 

The  Value  of  a  Fraction  is  the  quotient  of  the  numerator  divided 
by  the  denominator. 

Fractions  are  divided  into  two  classes  —  Common  and  Decimal 

A  Common  Fraction  is  one  in  which  the  unit  is  divided  into  any 
number  of  equal  parts. 

A  common  fraction  is  expressed  by  writing  the  numerator  above 
the  denominator  with  a  dividing  line  between. 


Viii  SUPPLEMENT 

Common  fractions  consist  of  three  principal  classes  —  Simple, 
Compound,  and  Complex, 

A  Simple  Fraction  is  one  whose  terms  are  whole  numbers. 

A  Proper  Fraction  is  a  simple  fraction  whose  numerator  is  less 
than  its  denominator. 

An  Improper  Fraction  is  a  simple  fraction  whose  numerator 
equals  or  exceeds  its  denominator. 

A  Compound  Fraction  is  a  fraction  of  a  fraction. 

A  Complex  Fraction  is  one  having  a  fraction  in  its  numerator,  or 
in  its  denominator,  or  in  both. 

A  Mixed  Number  is  a  whole  number  and  a  fraction  written 
together. 

The  Eeciprocal  of  a  Number  is  one  divided  by  that  number. 

The  Eeciprocal  of  a  Fraction  is  one  divided  by  the  fraction,  or 
the  fraction  inverted. 

PRINCIPLES.  —  Multiplying  the  numerator  or  dividing  the  de- 
nominator multiplies  the  fraction. 

Dividing  the  numerator  or  multiplying  the  denominator  divides 
the  fraction. 

Multiplying  or  dividing  both  terms  of  a  fraction  by  the  same 
number  does  not  alter  the  value  of  the  fraction. 

Eeduction  of  fractions  is  changing  their  terms  without  altering 
their  value. 

To  reduce  a  Fraction  to  Higher  Terms, 

RULE.  —  Multiply  both  numerator  and  denominator  by  the  same 
number. 

To  reduce  a  Fraction  to  its  Lowest  Terms, 

RULE.  —  Divide  both  terms  of  the  fraction  by  their  greatest 
common  divisor. 

A  fraction  is  in  its  lowest  terms  when  the  numerator  and  the 
denominator  are  prime  to  each  other. 


DEFINITIONS,    PRINCIPLES,   AND    RULES  IX 

To  reduce  a  Mixed  Number  to  an  Improper  Fraction. 

RULE.  —  Multiply  the  whole  number  by  the  denominator;  to  the 
product  add  the  numerator ;  and  place  the  sum  over  the  denom- 
inator. 

To  reduce  an  Improper  Fraction  to  a  Whole  or  to  a  Mixed  Number, 

RULE.  —  Divide  the  numerator  by  the  denominator. 

A  Oommon  Denominator  is  a  denominator  common  to  two  or 
more  fractions. 

The  Least  Oommon  Denominator  is  the  smallest  denominator 
common  to  two  or  more  fractions. 

To  reduce  Fractions  to  their  Least  Oommon  Denominator. 

RULE.  —  Find  the  least  common  multiple  of  all  the  denomi- 
nators for  the  least  common  denominator.  Divide  this  multiple  by 
the  denominator  of  each  fraction,  and  multiply  the  numerator  by 
the  quotient. 

ADDITION  OF  FRACTIONS 

PRINCIPLE.  —  Only  like  fractions  can  be  added. 

RULE.  — Reduce  the  fractions,  if  necessary,  to  a  common  denom- 
inator, and  over  it  write  the  sum  of  the  numerators. 

If  there  are  mixed  numbers,  add  the  fractions  and  the  whole 
numbers  separately,  and  unite  the  results. 

SUBTRACTION  OF  FRACTIONS 

PRINCIPLE.  —  Only  like  fractions  can  be  subtracted. 

RULE.  —  Reduce  the  fractions,  if  necessary,  to  a  common  denom- 
inator, and  over  it  write  the  difference  between  the  numerators. 

If  there  are  mixed  numbers  subtract  the  fractions  and  the  whole 
numbers  separately,  and  unite  the  results. 

MULTIPLICATION  OF  FRACTIONS 

RULE.  —  Reduce  whole  and  mixed  numbers  to  improper  frac- 
tions ;  cancel  the  factors  common  to  numerators  and  denomina- 
tors, and  write  the  product  of  the  remaining  factors  in  the  numer- 
ators over  the  product  of  the  remaining  factors  in  the  denominators. 


SUPPLEMENT 


DIVISION   OF  FRACTIONS 

EULES.  —  I.  Reduce  whole  and  mixed  numbers  to  improper 
fractions.  Reduce  the  fractions  to  a  common  denominator.  Divide 
the  numerator  of  the  dividend  by  the  numerator  of  the  divisor. 

II.  Invert  the  divisor  and  proceed  as  in  multiplication  of  frac- 
tions. 

To  reduce  a  Complex  Fraction  to  a  Simple  One. 

KULES.  —  I.  Multiply  the  numerator  of  the  complex  fraction 
by  its  denominator  inverted. 

II.  Multiply  both  terms  by  the  least  common  -multiple  of  the 
denominators. 

DECIMALS 

A  Decimal  Fraction  is  one  in  which  the  unit  is  divided  into 
tenths,  hundredths,  thousandths,  etc. 

A  Decimal  is  a  decimal  fraction  whose  denomination  is  indi- 
cated by  the  number  of  places  at  the  right  of  the  decimal  point. 

The  Decimal  Point  is  the  mark  used  to  locate  units. 

A  Mixed  Decimal  is  a  whole  number  and  a  decimal  written 
together. 

A  Complex  Decimal  is  a  decimal  with  a  common  fraction 
written  at  its  right. 

To  write  Decimals. 

RULE.  —  Write  the  numerator ;  and  from  the  right,  point  off  as 
many  decimal  places  as  there  are  ciphers  in  the  denominator, 
prefixing  ciphers,  if  necessary,  to  'make  the  required  number. 

To  read  Decimals. 

RULE.  —  Read  the  numerator,  and  give  the  name  of  the  right- 
hand  order. 

PRINCIPLES.  —  Prefixing  ciphers  to  a  decimal  diminishes  its 
value. 


DEFINITIONS,    PRINCIPLES,   AND   RULES  xi 

Removing  ciphers  from  the  left  of  a  decimal  increases  its  value. 
Annexing  ciphers  to  a  decimal  or  removing  ciphers  from  its 
right,  does  not  alter  its  value. 

To  reduce  a  Decimal  to  a  Common  Fraction. 

RULE.  —  Write  the  figures  of  the  decimal  for  the  numerator,  and 
1,  with  as  many  ciphers  as  there  are  places  in  the  decimal,  for  the  t 
denominator,  and  reduce  the  fraction  to  its  lowest  terms. 

To  reduce  a  Common  Fraction  to  a  Decimal, 

RULE.  —  Annex  decimal  ciphers  to  the  numerator,  and  divide  it 
by  the  denominator. 

To  reduce  Decimals  to  a  Common  Denominator. 

RULE.  —  Make  their  decimal  places  equal  by  annexing  ciphers. 

ADDITION  AND  SUBTRACTION  OF  DECIMALS 
Decimals  are  added  and  subtracted  the  same  as  whole  numbers. 

MULTIPLICATION  OF  DECIMALS 

RULE.  —  Multiply  as  in  whole  numbers,  and  from  the  right  of 
the  product,  point  off  as  many  decimal  places  as  there  are  decimal 
places  in  both  factors. 

DIVISION  OF  DECIMALS 

RULE.  —  Make  the  divisor  a  whole  number  by  removing  the 
decimal  point,  and  make  a  corresponding  change  in  the  dividend. 
Divide  as  in  whole  numbers,  and  place  the  decimal  point  in  the 
quotient  under  (or  over)  the  new  decimal  point  in  the  dividend. 

ACCOUNTS  AND  BILLS 

A  Debtor  is  a  person  who  owes  another. 

A  Creditor  is  a  person  to  whom  a  debt  is  due. 


Xii  SUPPLEMENT 

An  Account  is  a  record  of  debits  and  credits  between  persons 
doing  business. 

The  Balance  of  an  account  is  the  difference  between  the  debit 
and  credit  sides. 

A  Bill  is  a  written  statement  of  an  account. 

An  Invoice  is  a  written  statement  of  items,  sent  with  merchan- 
dise. . 

A  Eeceipt  is  a  written  acknowledgment  of  the  payment  of 
part  or  all  of  a  debt. 

A  bill  is  receipted  when  the  words,  "  Received  Payment,"  are 
written  at  the  bottom,  signed  by  the  creditor,  or  by  some  person 
duly  authorized. 

DENOMINATE   NUMBERS 

A  Measure  is  a  standard  established  by  law  or  custom,  by 
which  distance, 'capacity,  surface,  time,  or  weight  is  determined. 

A  Denominate  Unit  is  a  unit  of  measure. 

A  Denominate  Number  is  a  denominate  unit  or  a  collection  of 
denominate  units. 

A  Simple  Denominate  Number  consists  of  denominate  units  of 
one  kind. 

A  Compound  Denominate  Number  consists  oi  denominate  units  of 
two  or  more  kinds. 

A  Denominate  Praction  is  a  fraction  of  a  denominate  number. 

A  denominate  fraction  may  be  either  common  or  decimal, 

Reduction  of  denominate  numbers  is  changing  them  from  one 
denomination  to  another  without  altering  their  value. 

Reduction  Descending  is  changing  a  denominate  number  to  one 
of  a  lower  denomination. 

RULE.  —  Multiply  the  highest  denomination  by  the  number  re- 
quired to  reduce  it  to  the  next  lower  denomination,  and  to  the  prod- 
uct add  the  units  of  that  lower  denomination,  if  any.  Proceed 
in  this  manner  until  the  required  denomination  is  reached. 


DEFINITIONS,   PRINCIPLES,   AND   RULES  xiii 

Beduction  Ascending  is  changing  a  denominate  number  to  one  of 
a  higher  denomination. 

RULE.  —  Divide  the  given  denomination  successively  by  the 
numbers  that  will  reduce  it  to  the  required  denomination.  To  this 
quotient  annex  the  several  remainders. 

To  find  the  Time  between  Dates. 

RULE.  —  When  the  time  is  less  than  one  year,  find  the  exact 
number  of  days;  if  greater  than  one  year,  find  the  time  by  com- 
pound subtraction,  taking  30  days  to  the  month. 

PERCENTAGE 

Per  Cent  means  hundredths. 

Percentage  is  computing  by  hundredths. 

The  elements  involved  in  percentage  are  the  Base,  Bate,  Per- 
centage, Amount,  and  Difference. 

The  Base  is  the  number  of  which  a  number  of  hundredths  is 
taken. 

The  Bate  indicates  the  number  of  hundredths  to  be  taken. 

The  Percentage  is  one  or  more  hundredths  of  the  base. 

The  Amount  is  the  base  increased  by  the  percentage. 

The  Difference  is  the  base  diminished  by  the  percentage. 

To  find  the  Percentage  when  the  Base  and  Bate  are  Given, 
RULE.  —  Multiply  the  base  by  the  rate  expressed  as  hundredths. 
To  find  the  Bate  when  the  Percentage  and  Base  are  Given. 
RULE.  —  Divide  the  percentage  by  the  base. 
To  find  the  Base  when  the  Percentage  and  Bate  are  Given. 
RULE.  —  Divide  the  percentage  by  the  rate  expressed  as  hun- 
dredths. 

To  find  the  Base  when  the  Amount  and  Bate  are  Given. 
RULE.  —  Divide  the  amount  by  1  -j-  the  rate  expressed  as  hun- 
dredths. 


XIV  SUPPLEMENT 

To  find  the  Base  when  the  Difference  and  Kate  are  Given, 
RULE.  —  Divide  the  difference  by  I  —  the  rate  expressed  as  hun- 
dredths. 

PROFIT   AND   LOSS 

Profit  or  Loss  is  the  difference  between  the  buying  and  selling 
prices. 

In  Profit  and  Loss, 

The  buying  price,  or  cost,  is  the  base. 
The  rate  per  cent  profit  or  loss  is  the  rate. 
The  profit  or  loss  is  the  percentage. 

The  selling  price  is  the  amount  or  difference,  according  as  it 
is  more  or  less  than  the  buying  price. 

COMMERCIAL  DISCOUNT 

Commercial  Discount  is  a  percentage  deducted  from  the  list 
price  of  goods,  the  face  of  a  bill,  etc. 

The  Net  Price  of  goods  is  the  sum  received  for  them. 

In  Commercial  Discount, 
The  list  price,  or 


The  face  of  the  bill   '&•*«* 


• 

The  rate  per  cent  discount  is  the  rate. 

The  discount  is  the  percentage. 

The  list  price  diminished  by  the  discount  is  the  difference. 

In  successive  discounts,  the  first  discount  is  made  from  the  list 
price  or  the  face  of  the  bill ;  the  second  discount,  from  the  list 
price  or  face  of  the  bill  diminished  by  the  first  discount ;  and  so 
on. 

COMMISSION 

Commission  is  a  percentage  allowed  an 'agent  for  his  services. 
A  Commission  Agent  is  one  who  transacts  business  on   com- 
mission. 


>  ig 
) 


DEFINITIONS,    PRINCIPLES,   AND   RULES  XV 

A  Consignment  is  the  merchandise  forwarded  to  a  commission 
agent. 

The  Consignor  is  the  person  who  sends  the  merchandise. 

The  Consignee  is  the  person  to  whom  the  merchandise  is  sent. 

The  Net  Proceeds  is  the  sum  remaining  after  all  charges  have 
been  deducted. 

In  buying,  the  commission  is  a  percentage  of  the  buying  price; 
in  selling,  a  percentage  of  the  selling  price;  in  collecting,  a  per- 
centage of  the  sum  collected;  hence  : 

The  sum  invested,  or 

The  sum  collected 

The  rate  per  cent  commission  is  the  rate. 

The  commission  is  the  percentage. 

The  sum  invested  increased  by  the  commission  is  the  amount. 

The  sum  collected  diminished  by  the  commission  is  the  differ- 
ence. 

INSURANCE 

Insurance  is  a  contract  of  indemnity. 

Insurance  is  of  three  kinds  —  Fire,  Marine,  and  Life, 

Fire  Insurance  is  indemnity  against  loss  of  property  by  fire. 

Marine  Insurance  is  indemnity  against  loss  of  property  by  the 
casualties  of  navigation. 

Life  Insurance  is  indemnity  against  loss  of  life. 

The  Insurance  Policy  is  the  contract  setting  forth  the  liability  of 
the  insurer. 

The  Policy  Face  is  the  amount  of  insurance. 

The  Premium  is  the  price  paid  for  insurance. 

The  Insurer,  or  Underwriter,  is  the  company  issuing  the  policy. 

The  Insured  is  the  person  for  whose  benefit  the  policy  is  issued. 

In  Insurance, 

The  policy  face  is  the  base. 

The  rate  per  cent  premium  is  the  rate. 

The  premium  is  the  percentage. 


XVi  SUPPLEMENT 

TAXES 

A  Tax  is  a  sum  of  money  levied  on  persons  or  property  foi 
public  purposes. 

A  Personal,  or  Poll  Tax,  is  a  tax  on  the  person. 

A  Property  Tax  is  a  tax  of  a  certain  per  cent  on  the  assessed 
value  of  property. 

Property  may  be  either  personal  or  real. 

Personal  Property  consists  of  such  things  as  are  movable. 

Eeal  Property  is  that  which  is  fixed,  or  immovable. 

In  Taxes, 

The  assessed  value  is  the  base* 
The  rate  of  taxation  is  the  rate. 
The  tax  is  the  percentage. 

DUTIES 

Duties  are  taxes  on  imported  goods. 
Duties  are  either  Specific  or  Ad  Valorem. 
A  Specific  Duty  is  a  tax  on  goods  without  regard  to  cost. 
An  Ad  Valorem  duty  is  a  tax  of  a  certain  per  cent  on  the  cost 
of  goods. 

In  Ad  Valorem  Duties, 
The  cost  of  the  goods  is  the  base. 
The  rate  per  cent  duty  is  the  rate. 
The  ad  valorem  duty  is  the  percentage. 

INTEREST 

Interest  is  the  sum  paid  for  the  use  of  money. 

The  Principal  is  the  sum  loaned. 

The  Amount  is  the  sum  of  the  principal  and  interest. 

The  Kate  of  Interest  is  the  rate  per  cent  for  one  year. 

The  Legal  Eate  is  the  rate  fixed  by  law. 

Usury  is  interest  at  a  higher  rate  than  that  fixed  by  law. 

Simple  Interest  is  interest  on  the  principal  only. 


DEFINITIONS,    PRINCIPLES,   AND   RULES  IV11 

To  find  the  Interest  when  the  Principal,  Time,  and  Rate  are  Given. 

RULE.  —  Multiply  the  principal  by  the  rate  expressed  as  hun- 
dredths,  and  this  product  by  the  time  expressed  in  years. 

7  To  find  the  Time  when  the  Principal,  Interest,  and  Kate  are  Given. 

RULE.  —  Divide  the  given  interest  by  the  interest  for  one  year. 
^ 

To  find  the  Bate  when  the  Principal,  Interest,  and  Time  are  Given. 

RULE.  —  Divide  the  given  interest  by  the  interest  at  one  per 
cent. 
iV 

To  find  the  Principal  when  the  Interest,  Kate,  and  Time  are  Given. 

RULE.  —  Divide  the  given  interest  by  the  interest  on  $  1. 

To  find  the  Principal  when  the  Amount  and  Time  and  Kate  are 
Given. 

RULE.  —  Divide  the  given  amount  by  the  amount  of  $  1. 

INTEREST  BY  ALIQUOT  PARTS. 

To  find  the  Interest  for  Years,  Months,  and  Days. 

RULE.  —  Find  the  interest  for  one  year  and  take  this  as  many 
times  as  there  are  years. 

Take  the  greatest  number  of  the  given  months  that  equals  an 
aliquot  part  of  a  year  and  find  the  interest  for  this  time.  Take 
aliquot  parts  of  this  for  the  remaining  months. 

In  the  same  manner  find  the  interest  for  the  days. 

The  sum  of  these  interests  will  be  the  interest  required. 

To  find  the  Interest  when  the  Time  is  Less  than  a  Year. 

RULE. —  Find  the  interest  for  the  time  in  months  or  days  that 
will  gain  one  per  cent  of  the  principal. 

Find  by  aliquot  parts,  as  in  the  first  rule,  the  interest  for  the 
remaining  time. 

The  sum  of  these  interests  will  be  the  interest  required. 


XVlli  SUPPLEMENT 

INTEREST  BY  Six  PER  CENT  METHOD. 

To  find  the  Interest  at  6%, 

KULE.  —  Por  Tears:  Multiply  the  principal  by  the  rate  ex- 
pressed as  hundredths,  and  that  product  by.  the  number  of  years. 

For  Months :  Move  the  decimal  point  two  places  to  the  left,  and 
multiply  by  one-half  the  number  of  months. 

For  Days;  Move  the  decimal  point  three  places  to  the  left,  and 
multiply  by  one-sixth  the  number  of  days. 

To  find  the  interest  at  any  other  rate  per  cent,  divide  the  in- 
terest at  6%  by  6,  and  multiply  the  quotient  by  the  given  rate. 

To  find  Exact  Interest. 

RULE.  —  Multiply  the  principal  by  the  rate  expressed  as  hun- 
dredths,  and  that  product  by  the  time  expressed  in  years  of  365 
days. 

ANNUAL   INTEREST 

Annual  Interest  is  interest  payable  annually.  If  not  paid  when 
due,  annual  interest  draws  simple  interest. 

To  find  the  Amount  Due  on  a  Note  with  Annual  Interest,  when  the 
Interest  has  not  been  Faid  Annually, 

RULE.  —  Find  the  interest  on  the  principal  for  the  entire  time, 
and  on  each  annual  interest  for  the  time  it  remained  unpaid. 
The  sum  of  the  principal  and  all  the  interest  is  the  amount  due. 

COMPOUND   INTEREST 

Compound  Interest  is  interest  on  the  principal  and  on  the  un- 
paid interest,  which  is  added  to  the  principal  at  regular  inter- 
vals. The  interest  may  be  compounded  annually,  semi-annually, 
quarterly,  etc.,  .according  to  agreement. 

To  find  Compound  Interest, 

RULE.  —  Find  the  amount  of  the  given  principal  for  the  first 
period.  Considering  this  as  a  new  principal,  find  the  amount  of 


DEFINITIONS,    PRINCIPLES,    AND    RULES  xix 

it  for  the  next  period,  continuing  in  this  manner  for  the  given 
time. 

Find  the  difference  between  the  last  amount  and  the  given 
principal,  which  will  be  the  compound  interest. 

PARTIAL  PAYMENTS 

Partial  Payments  are  part  payments  of  a  note  or  debt.  Each 
payment  is  recorded  on  the  back  of  the  note  or  the  written 
obligation. 

UNITED  STATES  RULE.  —  Find  the  amount  of  the  principal  to 
the  time  when  the  payment  or  the  sum,  of  two  or  more  payments 
equals  or  exceeds  the  interest. 

From  this  amount  deduct  the  payment  or  sum  of  payments. 

Use  the  balance  then  due  as  a  new  principal,  and  proceed  as 
before. 

MERCHANTS'  RULE.  —  Find  the  amount  of  an  interest-bearing 
note  at  the  time  of  settlement. 

Find  the  amount  of  each  credit  from  its  time  of  payment  to  the 
~time  of  settlement ;  subtract  their  sum  from  the  amount  of  the 
principal. 

BANK  DISCOUNT 

Bank  Discount  is  a  percentage  retained  by  a  bank  for  advanc- 
ing money  on  a  note  before  it  is  due. 

The  Sum  Discounted  is  the  face  of  the  note,  or  if  interest-bear- 
ing, the  amount  of  the  note  at  maturity. 

The  Term  of  Discount  is  the  number  of  days  from  the  day  of 
discount  to  the  day  of  maturity. 

The  Bank  Discount  is  the  interest  on  the  sum  discounted  for 
the  term  of  discount. 

The  Proceeds  of  a  note  is  the  sum  discounted  less  the  bank  dis- 
count. 

Problems  in  bank  discount  are  calculated  as  problems  in 
interest. 


XX  SUPPLEMENT 

In  Bank  Discount, 

The  sum  discounted  is  the  principal. 

The  rate  of  discount  is  the  rate  of  interest. 

The  term  of  discount  is  the  time. 

The  bank  discount  is  the  proceeds. 

EXCHANGE 

Exchange  is  making  payments  at  a  distance  by  means  of  drafts 
or  bills  of  exchange. 

Domestic  Exchange  is  exchange  between  places  in  the  same 
country. 

Foreign  Exchange  is  exchange  between  different  countries. 

Exchange  is  at  par  when  a  draft,  or  bill,  sells  for  its  face 
value  ;  at  a  premium  when  it  sells  for  more  than  its  face  value  ; 
at  a  discount  when  it  sells  for  less. 

The  cost  of  a  sight  draft  is  the  face  of  the  draft  increased  by 
the  premium,  or  diminished  by  the  discount. 

The  cost  of  a  time  draft  is  the  face  of  the  draft  increased  by 
the  premium,  or  diminished  by  the  discount,  and  this  result 
diminished  by  the  bank  discount. 

To  find  the  Cost  of  a  Draft. 

RULE. —  Find  the  cost  of  $\  of  the  draft;  multiply  this  by  the 
face  of  the  draft. 

To  find  the  Pace  of  a  Draft. 

RULE.  —  Divide  the  cost  of  the  draft  by  the  cost  of  f  1  of  the 
draft. 

EQUATION  OF  PAYMENTS 

Equation  of  Payments  is  a  method  of  ascertaining  at  what  time 
several  debts  due  at  different  times  may  be  settled  by  a  single 
payment. 

The  Equated  Time  of  payment  is  the  time  when  the  several 
debts  may  be  equitably  settled  by  one  payment. 

The  Term  of  Credit  is  the  time  the  debt  has  to  run  before  it 
becomes  due. 


DEFINITIONS,   PKINCIPLES,   AND   RULES  XXi 

The  Average  Term  of  Credit  is  the  time  the  debts  due  at  different 
times  have  to  run,  before  they  may  be  equitably  settled  by  one 
payment. 

To  find  the  Equated  Time  of  Payment  when  the  Terms  of  Credit 
begin  at  the  Same  Date. 

RULE.  —  Multiply  each  debt  by  its  term  of  credit,  and  divide  the 
sum  of  the  products  by  the  sum  of  the  debts.  The  quotient  will  be 
the  average  term  of  credit. 

Add  the  average  term  of  credit  to  the  date  of  the  debts,  and  the 
result  will  be  the  equated  time  of  payment. 

To  find  the  Equated  Time  when  the  Terms  of  Credit  begin  at 
Different  Dates. 

RULE.  —  Find  the  date  at  which  each  debt  becomes  due.  Select 
the  earliest  date  as  a  standard. 

Multiply  each  debt  by  the  number  of  days  between  the  standard 
date  and  the  date  when  the  debt  becomes  due,  and  divide  the  sum, 
of  the  products  by  the  sum  of  the  debts.  The  quotient  will  be  the 
average  term  of  credit  from  the  standard  date. 

Add  the  average  term  of  credit  to  the  standard  date,  and  the 
result  will  be  the  equated  time  of  payment. 

RATIO 

Ratio  is  the  relation  one  number  bears  to  another  of  the  same 
kind. 

The  Terms  of  the  ratio  are  the  numbers  compared. 

The  Antecedent  is  the  first  term. 

The  Consequent  is  the  second  term. 

The  antecedent  and  consequent  form  a  couplet. 

PRINCIPLES. — See  Fractions. 

PROPORTION 

A  Proportion  is  formed  by  two  equal  ratios. 

The  Extremes  of  a  proportion  are  the  first  and  last  terms. 

The  Means  of  a  proportion  are  the  second  and  third  terms. 


XX11  SUPPLEMENT 

PEINCIPLES.  —  The  product  of  the  means  is  equal  to  the  prod- 
uct of  the  extremes. 

Either  mean  equals  the  product  of  the  extremes  divided  by  the 
other  mean. 

Either  extreme  equals  the  product  of  the  means  divided  by  the 
other  extreme. 

RULE  FOR  PROPORTION.  —  Represent  the  required  term  by  x. 

Arrange  the  terms  so  that  the  required  term  and  the  similar 
known  term  may  form  one  couplet,  the  remaining  terms  the  other. 

If  the  required  term  is  in  the  extremes,  divide  the  product  of  the 
means  by  the  given  extreme. 

If  the  required  term  is  in  the  means,  divide  the  product  of  the 
extremes  by  the  given  mean. 

PARTNERSHIP 

Partnership  is  an  association  of  two  or  more  persons  for  busi- 
ness purposes. 

The  Partners  are  the  persons  associated. 

The  Capital  is  that  which  is  invested  in  the  business. 

The  Assets  are  the  partnership  property. 

The  Liabilities  are  the  partnership  debts. 

To  find  the  Profit,  or  Loss,  of  Each  Partner  when  the  Capital  of 
Each  is  Employed  for  the  Same  Period  of  Time, 

RULE.  —  Find  the  part  of  the  entire  profit,  or  loss,  that  each 
partner  s  capital  is  of  the  entire  capital. 

To  find  the  Profit,  or  Loss,  of  Each  Partner  when  the  Capital  of 
Each  is  Employed  for  Different  Periods  of  Time. 

RULE.  —  Find  each  partner  s  capital  for  one  month,  by  multi- 
plying the  amount  he  invests  by  the  number  of  months  it  is 
employed;  then  find  the  part  of  the  entire  profit,  or  loss,  that  each 
partner  s  capital  for  one  month  is  of  the  entire  capital  for  one 
month. 


DEFINITIONS,    PEINCIPLES,    AND   RULES 


INVOLUTION 

A  Power  of  a  number  is  the  product  obtained  by  using  that 
number  a  certain  number  of  times  as  a  factor. 

The  First  Power  of  a  number  is  the  number  itself. 

The  Second  Power  of  a  number,  or  the  Square,  is  the  product  of 
a  number  taken  twice  as  a  factor. 

The  Third  Power  of  a  number,  or  the  Cube,  is  the  product  of  a 
number  taken  three  times  as  a  factor. 

An  Exponent  is  a  small  figure  written  a  little  to  the  right  of  the 
upper  part  of  a  number  to  indicate  the  power. 

Involution  is  finding  any  power  of  a  number. 

To  find  the  Power  of  a  Number, 

RULE.  —  Take  the  number  as  a  factor  as  many  times  as  there 
are  units  in  the  exponent. 

EVOLUTION 

A  Boot  is  one  of  the  equal  factors  of  a  number. 

The  Square  Boot  of  a  number  is  one  of  its  two  equal  factors. 

The  Cube  Boot  of  a  number  is  one  of  its  three  equal  factors. 

Evolution  is  finding  any  root  of  a  number. 

Evolution  may  be  indicated  in  two  ways:  by  the  Radical 
Sign,  V~,  or  by  a.  fractional  exponent. 

The  Index  of  a  root  is  a  small  figure  placed  a  little  to  the  left 
of  the  upper  part  of  the  radical  sign,  to  indicate  what  root  is  to 
be  found.  In  expressing  square  root,  the  index  is  omitted. 

In  the  fractional  exponent,  the  numerator  indicates  the  power 
to  which  the  number  is  to  be  raised  ;  the  denominator  indicates 
the  root  to  be  taken  of  the  number  thus  raised. 

To  find  the  Square  Boot  of  a  Number, 

RULE.  —  Point  off  in  periods  of  two  figures,  commencing  at 
units.  Find  the  greatest  square  in  the  first  period  and  place  the 
root  in  the  quotient.  Subtract  this  square  from  the  first  period, 
and  bring  down  the  next  period. 


xxiv  SUPPLEMENT 

Multiply  the  quotient  figure  by  two,  and  use  it  as  a  trial  divisor. 
Place  the  second  figure  in  the  quotient,  and  annex  it  also  to  the 
trial  divisor.  Then  multiply  the  figures  in  the  trial  divisor  by  the 
second  quotient  figure,  and  subtract. 

Bring  down  the  next  period,  and  proceed  as  before  until  the 
square  root  is  found. 

To  find  the  Square  Eoot  of  a  Traction. 

RULE.  —  Reduce  the  fraction  to  its  simplest  form,  and  find  the 
square  root  of  each  term  separately. 

To  find  the  Cube  Eoot  of  a  Number. 

RULE.  —  Point  off  in  periods  of  three  figures  each,  beginning  at 
units. 

Find  the  greatest  cube  in  the  first  period  and  place  the  root  in 
the  quotient.  Subtract  this  cube  from  the  first  period,  and  bring 
down  the  next  period. 

Multiply  the  square  of  the  first  quotient  figure  by  three  and 
annex  two  ciphers  for  a  trial  divisor.  Place  the  second  figure  in 
the  quotient.  Then,  to  the  trial  divisor  add  three  times  the  prod- 
uct of  the  first  and  second  figures,  also  the  square  of  the  second. 
Multiply  this  sum  by  the  second  figure  and  subtract. 

Bring  down  the  next  period,  and  proceed  as  before  until  the  cube 
root  is  found. 

To  find  the  Cube  Boot  of  a  Traction. 

RULE.  —  Reduce  the  fraction  to  its  simplest  form,  and  find  the 
cube  root  of  each  term  separately. 

STOCKS  AND   BONDS. 

Capital  Stock  is  the  money  or  property  employed  by  a  corpora- 
tion in  its  business. 

A  Share  is  one  of  the  equal  divisions  of  capital  stock. 

The  Stockholders  are  the  owners  of  the  capital  stock. 

The  Par  Value  of  stock  is  the  face  value. 

The  Market  Value  of  stock  is  the  sum  for  which  it  may  be  sold. 


DEFINITIONS,   PRINCIPLES,   AND   RULES  XXV 

Stock  is  at  a  premium  when  the  market  value  is  above  the 
par  value ;  at  a  discount,  when  below  par. 

Bonds  are  interest-bearing  notes  issued  by  a  government  or  a 
corporation. 

A  Dividend  is  a  percentage  apportioned  among  the  stockholders. 
A  Stock  Broker  is  a  person  who  deals  in  stocks. 
Brokerage  is  a  percentage  allowed  a  stock  broker  for  his  services. 
In  Stocks  and  Bonds, 

The  par  value  is  the  base. 

Th«  rate  per  cent  premium,  or  discount,  is  the  rate. 

The  premium,      "j 

discount,  or  >  is  the  percentage. 
dividend       J 

amount,  or 
difference. 


The  market  value  is  the  j 


NOTES,  DRAFTS,  AND  CHECKS. 

A  Promissory  Note  is  a  written  promise  to  pay  a  specified  sum 
on  demand,  or  at  a  specified  time. 

The  Face  of  a  note  is  the  sum  named  in  the  note. 

The  Maker  is  the  person  who  signs  it. 

The  Payee  is  the  person  to  whom  the  sum  specified  is  to  be 
paid. 

The  Indorser  is  the  person  who  signs  his  name  on  the  back  of 
the  note,  thus  becoming  liable  for  its  payment  in  case  of  default 
of  the  maker. 

An  Interest-bearing  Note  is  one  payable  with  interest. 

If  the  words  "  with  interest "  are  omitted,  interest  cannot  be 
collected  until  after  maturity. 

A  Demand  Note  is  one  payable  when  demand  of  payment  is 
made. 

A  Time  Note  is  one  payable  at  a  specified  time. 

A  Joint  Note  is  one  signed  by  two  or  more  persons  who  jointly 
promise  to  pay. 


XXVI  SUPPLEMENT 

A  Joint  and  Several  Note  is  one  signed  by  two  or  more  persons 
who  jointly  and  severally  promise  to  pay. 

In  a  joint  note,  each  person  is  liable  for  the  whole  amount, 
but  they  must  all  be  sued  together.  In  the  joint  and  several 
note,  each  is  liable  for  the  whole  amount,  and  may  be  sued 
separately. 

A  Negotiable  Note  is  one  that  may  be  transferred  or  sold.  It 
contains  the  words  "  or  bearer,"  or  "  or  order." 

A  Non-negotiable  Note  is  one  not  payable  to  the  bearer,  nor  to 
the  payee's  order. 

The  Maturity  of  a  note  is  the  day  on  which  it  legally  falls  due. 

A  Draft,  or  Bill  of  Exchange,  is  a  written  order  directing  the 
payment  of  a  specified  sum  of  money. 

The  Face  of  a  draft  is  the  sum  named  in  it. 

The  Drawer  is  the  person  who  signs  the  draft. 

The  Drawee  is  the  person  ordered  to  pay  the  sum  specified. 

The  Payee  is  the  person  to  whom  the  sum  specified  is  to  be 
paid. 

A  Sight  Draft  is  one  payable  when  presented. 

A  Time  Draft  is  one  payable  at  a  specified  time. 

An  Acceptance  of  a  time  draft  is  an  agreement  by  the  drawee 
to  pay  the  draft  at  maturity,  which  he  signifies  by  writing  across 
the  face  of  the  draft  the  word  "  accepted  "  with  the  date  and  his 
name. 

A  Check  is  an  order  on  a  bank  or  banker  to  pay  a  specified 
sum  of  money.  * 


CALIf 


ANSWERS.  — PART  II. 


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Page  223. 

Page  218. 

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16.    16  marbles. 

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ANSWERS. 


14.   141 

18.  31,660,868. 

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Page  234. 

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4.  24. 

3.  561,276,891. 

50.   28,508$. 

5.  4. 

51.   69,763|. 

11.  5. 

Page  230. 

52.  598,686$. 

12.   15. 

Page  226. 

3.   12,642,968. 

53.   3,600,925f. 

13.   13. 

1.  $609,340.37. 

4.   8625. 

54.   21,436,213$. 

14.   31. 

2.  $680,494.41. 

5.   980,304. 

15.   5. 

3.  $840,200.33. 

6.  $439.11. 

16.   11. 

7.  $314.87. 

Page  232. 

17.  17. 

8.   7225. 

1.  $18,016.14. 

18.   25. 

Page  227. 

10.   $55,350. 

2.   1,058,213. 

19.   5. 

4.   350,879,581. 

11.   1,207,053. 

3.  $3405.78. 

20.  8. 

5.  627,020,401. 

13.   1376  yards. 

4.  8502. 

31.   16. 

6.   589,140,749. 

14.  998,392. 

5.  21,263,502. 

32.   8. 

7.   668,386,689. 

6.   747f. 

33.  20$. 

8.   777,993,982. 

7.  22,432$ff. 

34.  30$. 

9.   713,200,695. 

Page  231. 

35.   10. 

10.  578,616,033. 

25.   12|. 

26.   19£. 

Page  233. 

27.  42$, 

3.   19,656. 

Page  236. 

Page  228. 

28.   79$. 

4.  381. 

41.   24. 

11.  65,461,219. 

29.  80f. 

5.  $62.50. 

42.   40. 

12.   615,808,906. 

30.  128f. 

43.   17. 

13.  99,090,910. 

31.   101$. 

1.   674,022,122 

44.   61. 

14.   200,290,240. 

32.   169TV 

pieces. 

45.   9. 

15.   249,054. 

33.  215-&. 

2.  $2,466,338.49 

.  46.   90. 

16.   26,081. 

34.   177TV 

3.   38,788. 

47.   6. 

17.    102,900,999. 

35.   lllf. 

4.   $9332.86. 



ANSWERS. 


1.  $1283.35. 

5.  80,191. 

4.  $378.75. 

22.  9280  ounces. 

2.  $845.95. 

6.   74  acres. 

6.   41  days. 

23.   750  pounds. 

3.  $2121.75. 

7.  46,200^. 

6.  5578$  bu. 

24.   15cwt. 

4.  $857.62. 

8.   18  patterns. 

7.   148^  gal. 

25.  $1. 

5.  $1247.80. 

26.   f  ton. 

6.  $769.89. 

Page  239. 

Page  250. 

27.  4  da.  16  hr. 

7.  $530.25. 

9.  2582f|f 

8.   1,268,459,505 

28.   50  yards. 

8.  $853.58. 

10.  $89.60. 

pounds. 

29.  $8.25. 

9.  $1250.93. 

13.   5  quarts. 

9.   1,125,025,619 

30.   32  cents. 

10.  $712.50. 

14.   71  quarts. 

yards. 

31.  fcwt. 

15.   4  pecks. 

10.   30  hours. 

32.  348  pints. 

16.  9. 

12.  2,111,310,206. 

Page  237. 

17.  $2.10. 

13.   10,209,990 

Page  253. 

1.  $1115.02. 

18.  49,275. 

pieces. 

4.   39,312. 

2.  $505.44. 

14.  3684  quarts. 

24,568. 

3.  $1592.64. 

Page  246. 

15.  $1645.56. 

5.  152&. 

4.  $9263.05. 

1.  40  days. 

69fff. 

5.  $1526.25. 

2.   1050  yards. 

Page  251. 

6.  $967.20. 

3.   25  sheep. 

1.    180  hours. 

Page  258. 

7.  $7133.80. 

4.   149  pounds. 

2.    180  hours. 

1.  $134,083.44. 

8.  $1072.56. 

5.  300  bushels. 

3.   7200  seconds. 

2.  $108,350.78. 

9.  $23.76. 

6.   1402  pounds. 

4.  $12.75. 

3.  $27,437.70. 

10.  $58.24. 

7.  36  cents. 

6.  $2.48. 

4.  $56,284.66. 

8.   66|  cents. 

6.   40  pints. 

5.  $11,672.66. 

7.   160hf.pt. 

6.  $96,229.43. 

Page  238. 

Page  247. 

8.  20  packages. 

11.   $92.88. 

9.   522  rabbits. 

9.   25  cents. 

Page  259. 

12.  $989.90. 

10.  $2837.50. 

7.   1,000,342. 

13.  $102.52. 

11.  $11,159. 

Page  252. 

8.   854,822. 

14.  $133.38. 

12.   24  horses. 

10.   6  gallons. 

9.  3,649,094. 

15.  $140.60. 

13.   1000  bars. 

11.   18,000  sec. 

10.   11,810,804. 

16.  $34.98. 

12.  10,080  rain. 

11.   77,472,965. 

17.  $53.07. 

Page  249. 

13.   800  minutes. 

12.   33,845,968. 

18.  $103.60. 

5.  915f|i 

14.   512  quarts. 

13.  27,749,898. 

19.  $1591.62. 

6.   16  days. 

15.   757  ounces. 

14.   28,338,290f. 

20.  $4879.77. 

7.   19,208  Ib. 

16.  14  Ib.  13  oz. 

15.  32,136,750. 



8.  57  inches. 

17.  24  hr.  54  rain. 

16.   6.248.365J. 

1.  4,680,785f 

18.  1008  hours. 

17.  29,654,230. 

2.  2.478,208. 

1.  $86,362f»flV 

19.   744  hours. 

18.  63,257,616$. 

3.  $258,715,000. 

2.  1288  pieces. 

20.   98  inches. 

19.  857,375. 

4.  4919. 

3.  $566. 

21.   128,000  oz. 

20.   274,170. 

ANSWERS. 


21.   72,243. 

7.   $1536.65. 

15.   23.495. 

Page  273. 

22.   109,243,616. 

8.   49  T.  820  Ib. 

16.    18.168. 

1.   6.375  yards. 

23.   115,242f. 

9.   22. 

17.   2359.925. 

2.   $95.386. 

24.   16,610,750. 

10.   $80. 

18.    748.311. 

3.   2060g57  acres. 

11.  $1.25. 

19.    1062.556. 

4.   $14,000. 

Page  260. 

12.   25  cents. 

20.    799.511. 

5.   67  yards. 

25.   93/ft. 

13.   60  yards. 

26.  2175. 

Page  269. 

Page  274. 

27.   1025HI- 

Page  263. 

41.   1.08. 

6.   60  gills. 

OQ       OQ^  4138 
^    *              T^TTTo* 

14.   $22. 

42.   400.4. 

7.    12  bushels. 

29.   530H**. 

15.    10  papers. 

43.   780.8. 

8.   26TV  gallons. 

30.   6943.&VV 

44.    780.8. 

9.  $544. 

31.   5565Ty/^t. 

1.   $12.04. 

45.  2.68. 

10.  $1960. 

32.   15,168|ff|f. 

2.   $19.63. 

46.   1.536. 

13.  $4.50  gain. 

33,   2708. 

3.   $107.52. 

47.   1.536. 

14.  $1.76. 

34.   920ff±£. 

4.  |8.29. 

48.   55.272. 

15.   704  pints. 

35.   870ff7ff. 

49.   .485. 

16.  $679. 

36.   216^f£ff. 

Page  264. 

50.   4344. 

17.  $15.66  gain. 

5.   $122.75. 

51.   330. 

18.   $2.16. 

1.   39. 

52.   960. 

20.   8  marbles. 

2.   11. 

Page  266. 

53.    18. 

21.   90  cents. 

3.    12. 

1.   34,876f  av- 

54.  3801. 

22.   88  quarts. 

4.   88. 

erage    ap- 

55. 59.13. 

5.    181. 

plications. 

56.   725.56. 

Page  275. 

6.   77. 

$245,468.65^ 

57.  376.68. 

23.   90  cents. 

7.   690. 

average 

58.   334.508. 

24.  558  pupils. 

8.   17. 

surplus. 

59.   62,7. 

25.    110  feet. 

9.   11. 

5.   $4.45f. 

60.    1.8. 

26.   $1806. 

10.   123. 

6.   50T9^°T  cts. 

27.   20  cents. 

28.  $2.48. 

Page  261. 

Page  267. 

Page  271. 

29.   491  gills. 

11.    13. 

5.    122.995. 

31.   $241.25. 



12.   20. 

6.   293.056. 

32.    1968.5yd. 

4.   21«. 

7.   59.556. 

33.   35.4  pounds. 

5.   7f|. 

Page  262. 

8.   404.529. 

34.   1212.5  Ib. 

6.   95  cents. 

1.    $3. 

9.   390.732. 

35.   2.64  tons. 

2.  $3. 

10.   300.417. 

36.   $3364.02. 

Page  276. 

3.  $550. 

11.  480.507. 

37.   19  pints. 

7.  $1.25. 

4.   $1120. 

12.    939.186. 

38.   .125  peck. 

8.   22|  yards. 

5.  $105. 

13.    1180.106. 

39.  $58.50. 

9.   $80.50. 

6.   5  cents. 

14.   104.231. 

40.  $1751.56^. 

10.    1050  hours. 

ANSWERS. 


11.  33|  hours. 

17.   321|  sq.ft. 

21.  74. 

Page  283. 

13.  2J|  miles. 

18.   320  sq.  ft. 

22.  3ft. 

1.  f 

14.   $148.03. 

19.   400  sq.  ft. 

23.    16  Jf. 

2.  |J. 

15.   71  weeks. 

20.    150  sq.  ft. 

24.  318. 

3-  f- 

16.  $1.87J. 

25.  4ft. 

4.  f. 

17.  $2.02. 

Page  279. 

26.  22|. 

5.  |J. 

18.   254  hf.  pt. 

1.    10,937.5  sq. 

27.  46f. 

6.  I 

19.  9717ftf£. 

ft. 

28.   2437. 

7.  ff 

2.   2fsq.  yd. 

29.  507f 

8.   H- 

3.  4£  sq.  yd. 

30.   ft. 

9.  $. 

Page  277. 

4.   117  sq.  m. 

10.  f  . 

•  1.    182  sq.  in. 

5.   $15. 

11-  H- 

2.    153  sq.  in. 

6.  $48.75. 

Page  282. 

12.  f 

3.   126  sq.  in. 

7.  $2.40. 

1.  2,  43. 

4.   345  sq.  in. 

8.   672  sq.  rodf. 

2.   3,29. 

Page  284. 

13.   180  sq.  in. 

9.   30  sq.  yd. 

3.  2,  2,  2,  11. 

3.   f 

14.    144  sq.  in. 

10.  4  sq.  yd. 

4.   2,  3,  3,  5. 

4    14 

15.    192  eq.  in. 

5.   7,  13. 

5.  i 

16.   360  sq.  in. 

Page  280. 

6.   2,  2,  23. 

6-  A- 

17.   450  sq.  in. 

1.  22H- 

7.  3,31. 

7.  f 

18.    1419  sq.  in. 

2.   12*. 

8.   2,47 

8-  f 

19.    1180  sq.  in. 

3.   18|£J. 

9.  5,  19. 

9.   ft. 

20.   2205  sq.  in. 

4.  48f££. 

10.   2,  2,  2,  2,  2, 

10.  f. 

5.  47ft. 

3. 

11.  ft. 

Page  278. 

6.  90^. 

11.   2,  2,  5,  5. 

12.    f. 

1.    168  sq.  ft. 

7.  3ft. 

12.   2,  2,  2,  3,  5. 

13.  ft. 

2.   255  sq.  ft. 

8.  48*f. 

13.  2,  3,  5,  7. 

14.  f. 

3.   209  sq.  ft. 

9.   103$. 

14.  2,  2,  2,  2,  3, 

15.   ft. 

4.   345  sq.ft. 

10.  99J|. 

5. 

5.   288  sq.ft. 

15.  2,  2,  2,  3,  3, 

Page  286 

6.   348  sq.  ft. 

Page  281. 

5. 

1.   120. 

7.    186  sq.ft. 

11.  8|i. 

16.  2,  2,  2,  2,  2, 

2.   900. 

8.   186  sq.ft. 

12.  6Jf. 

2,  3,  3. 

3.   840. 

9.   300  sq.  ft. 

13.   38  J. 

17.  2,  2,  2,  3,  5, 

4.   240. 

10.   300  sq.ft. 

14.   17|f. 

7. 

5.  600. 

11.   423  sq.  ft. 

16.   19$. 

18.  2,  2,  2,  2,  2, 

6.  48. 

12.   444  sq.ft. 

16.   16H- 

2,  2,  3,  3. 

7.   360. 

13.   308  sq.ft. 

17.   24H- 

19.   2,  2,  2,  2,  2, 

8.   231. 

14.   426  sq.  ft. 

18.   22$$. 

2,  3,  3,  3. 

9.   720. 

15.   386  sq.ft. 

19.  6262Jf 

20.   2,  2,  2,  2,  2, 

10.   420. 

16.   843  sq.  ft. 

20.   199ff 

3,  3,  7. 



ANSWERS. 


1.    16ff 

8.    532  bags. 

2.   105ff. 

14.    3i 

2.   84H- 

Page  290. 

3.    117ff. 

15.   37f. 

3.    66  83- 

9.    52.272  acres. 

4.    121f|. 

16.   3. 

4.    31f. 

10.    26,100  sec. 

5.    219|f 

17.  42. 

5.    61A- 

11.   $6.35. 

6.    344AV 

18.    3ff. 

6.       2-js^jj. 

12.    $7.10. 

7.    213f|f. 

19.    J. 

7.    82tf. 

13.    20$  yards. 

8.    168fff. 

20.    i 

8.    101T%y7. 

14.    SOcts.     15.  |. 

9.   420T5T$:r§g-. 

21.    A- 

9.    41TVo. 

16.    68  marbles. 

10.   342A&. 

22.   A- 

10.    23^. 

17.    80  cents. 

23.    57|. 

18.   $119.46. 

Page  297. 

24.    41  1. 

Page  287. 

19.    58£  miles. 

3.    36  days. 

25.   47|f. 

11.    8A. 

20.  $6.90. 

4.    68  bags. 

26.    22A- 

12.   9A- 

5.    $2.99. 

27.   10J. 

•iq       QC  97 
J.O.     "^Tjj- 

Page  293. 

6.   $78. 

28.   9A- 

14.   74f|. 

1.    $581,812,541,- 

7.   $2844. 

29.    llf. 

15.    79ff. 

985.20. 

8.   $1.50. 

30.    15f. 

16.    223AV 

2.    3,417,600 

9.    $12. 

31.    3|. 

17.   471T6oV 

acres. 

10.   $147.60. 

32.   ||. 

18.    10T%6^. 

11.  $704. 

33.  ff. 

19.   43£f. 

Page  294. 

12.    36  days. 

34.   5A- 

20.    9Tj^. 

3.    $2480. 

OK          C    1  *7 

oD.     ^T?4' 

21.   A- 

6.    156  pounds. 

Page  298. 

36.    178|. 

22.   A- 

7.    $422. 

13.  $2.31. 

37.    19-^. 

23.   17A- 

8.    14,688  ounces. 

14.    60  days. 

38.    7ff. 

24.    $fj. 

10.   3  cents. 

15.   87^  cents. 

39.    2f. 

OK        101 

40      1  3 

/SO.       lo  .,  . 

26.    674£. 

1.   $4,532,088.68. 

Page  299. 

*"«    -I5(y- 

27.    1A. 

1.    64. 

Page  301. 

28.    3. 

Page  295. 

2.    96. 

1.    10. 

29.    1036f 

2.    2551  gal.  1  qt. 

3-   «. 

2.   f 

30.    &pfc. 

Ipt. 

4-   ff. 

3.    23£. 

3.   $25.74  lost. 

5.   £. 

4.    3|. 

Page  289. 

6.  $524,470,971.- 

6.    246. 

5.    A- 

1.    19|  cents. 

05. 

7.   436. 

i5' 

2.    75  cents. 

7.    $3.33£. 

8.    2221f 

7.    ^. 

3.    416^-  miles. 

8.    $2600. 

9.    3115A- 

8.    2f 

4.    2. 

9.    $1017.25. 

10.    15. 

9-   If- 

5.    $5. 

11.    9. 

10.  T4/T. 

6.   $107.48. 

Page  296. 

12.    H. 

11.  IA« 

7.   f. 

1.    135. 

13.    115. 

12.   |, 

ANSWERS. 


13.    If 

Page  307. 

Page  313. 

4.    27. 

14.    2|. 

1.   $2.43f 

4.  53  hours  20 

5.   72.96. 

15.  J. 

2.    1  ft.  3  in. 

minutes. 

6.    10.8. 

16.   A- 

3.    10^  pounds. 

5.   $4044.63. 

7.   81.666. 

17.   6. 

4.    y^. 

$2.36.-) 

8.   47.25. 

18.   *. 

5.    60  cents. 

$2.26.  !• 

9.    24. 

19.     TV 

6.   96  hours. 

$2.15.J 

10.    124.962. 

20.      T^T- 

7.   $1.14. 

6.   $5,400,000. 

21.    f 

8.    30  days. 

22.   A. 

9.   $76.25. 

3.   930,435,246 

Page  319. 

23.   3H- 

pieces. 

11.    112. 

24.    2^. 

Page  308. 

4.    10,209,990 

12.   335. 

25.    2|ff. 

10.   32  days. 

pieces. 

13.   445.9. 

26.    2^- 

11.    6  yards. 

7.   $320. 

14.    120. 

27.   Iff 

12.    1,260,000 

8.   $1.33^- 

15.    26.748. 

28.    iy&. 

cu.  ft. 

16.   372.6. 

29.    !&• 

13.   $7.50. 

Page  314. 

17.   473.184. 

30.    3f 

14.   90  cents. 

1.    29  pupils. 

18.   222. 

31.   6f 

15.   964  pounds. 

2.   I 

19.    10.92. 

32.   4f 

16.   60  cents. 

4.   $255.93|. 

20.    15,701.57. 

33.    17f 

17.   31^  cents. 

5.    8. 

21.    32. 

34.    17f 

18-   if 

6.    2ft. 

22.   3.2. 

35.    30Jf 

19.  $161. 

7-   f. 

23.   700. 

36.   3ojf 

20.   Twice  as  old. 

8.    I 

24.    40. 

37.   5f. 

21.   37$. 

9.   16^. 

25.    40. 

38.    33^. 

22.  $1430. 

10.   $45. 

26.   2. 

39.    37f 

23.    2  days. 

27.   3.2. 

40.   6|. 

24.   ffo. 

Page  315. 

28.   30.6. 

41.   101^. 

25.   $160. 

Carriages, 

29.   200. 

42.   2^,. 

2,698,526. 

30.   144. 

Page  311. 

Equestrians, 

31.   32,000. 

Page  305. 

1.    $161.85. 

132,137. 

32.    121. 

7.   If. 

2.   $27.36. 

Pedestrians, 

33.    13,500. 

8.  $1}}. 

3.   $35.96. 

13,730,597. 

34.    12.5. 

9.    lOf 

4.   $54.95. 

Total, 

35.    42.1. 

10.  4. 

16,567,956. 

36.    15,100. 

11.  59f 

Page  312. 

37.    1510. 

12.  lift. 

5.   $78.27. 

Page  318. 

38.    151. 

1Q        lift 

1.    80. 

AO.       *Sy* 

14.    1^. 

3.   $882,258.55. 

2.    80. 

Page  320. 

15.    4^,%. 

4.  $4,211,587.67. 

3.    28.8. 

39.    21. 

ANSWERS. 


40. 

21. 

4. 

420.168+ 

8. 

$28. 

39. 

19  Ib.  11  oz. 

41. 

21. 

marks. 

9. 

82  cents. 

40. 

61  yr.  11  mo. 

42. 

21. 

5. 

103.771. 

10. 

4  Ib.  10  oz. 

41. 

25  ft.  2  in. 

43. 

240. 

6. 

2699.73. 

11. 

6  bu.  1  pk. 

42. 

105  min.  57 

44. 

300. 

7. 

77.7. 

5  qt. 

sec. 

45. 

3.5. 

8. 

.3. 

12. 

4  gallons. 

43. 

50  years. 

46. 

12,360. 

9. 

1864  Ib. 

1^-  pints. 

44. 

35  wk.  2  da. 

47. 

122.5. 

10. 

62.832  in. 

13. 

$  3.05. 

45. 

13  miles. 

48. 

.016. 

14. 

9  bu.  1  qt. 

46. 

60  yards. 

49. 

400. 

3. 

61. 

47. 

50  gal.  2  qt. 

50. 

.007. 

4. 

540. 

48. 

13  pk.  2  qt. 

51. 

47. 

6. 

7f|.                   Page  328. 

49. 

74  bu.  1  pk. 

52. 

.064. 

.7. 

"Ht  pounds. 

15. 

63,360  in. 

50. 

50  quarts. 

53. 

13.5. 

16. 

924  feet. 

51. 

50  quarts. 

54. 

43647.89+. 

Page  324. 

17. 

14,080  rails. 

52. 

74  bu.  1  pk. 

55. 

2.384+. 

8. 

$11. 

18. 

3840  steps. 

53. 

48  pk.  3  qt. 

56. 

.264  +  . 

9. 

12  days. 

19. 

55  minutes. 

54. 

146  gallons. 

57. 

20.001  +  . 

10. 

14  yr.  1  mo. 

20. 

15hr.  17min. 

55. 

200  yards. 

58. 

4.405  +. 

11. 

6  miles. 

14hr.l8min. 

56. 

63  mi.   175 

59. 

24.8. 

12. 

$  1.92. 

21. 

276  ounces. 

rd. 

60. 

2.634+. 

13. 

87£. 

22. 

169,560  Ib. 

57. 

77  wk.  1  da. 

3rV 

23. 

151  quarts. 

58. 

72  yr.  6  mo. 

Page  321. 

14. 

$5.86. 

24. 

360  pints. 

59. 

81   min.    10 

4. 

.812. 

25. 

127  pints. 

sec. 

5. 

.105. 

Page  325. 

26. 

Ill  pecks. 

60. 

113  feet. 

6. 

1.06. 

2. 

5,026,101. 

27. 

391  quarts. 

61. 

6  ft.  5  in. 

7. 

.143. 

3. 

1,359,908. 

28. 

1344  pints. 

62. 

22  min.   43 

8. 

.025. 

4. 

9319fff. 

29. 

47,520  yd. 

sec. 

9. 

.044. 

5. 

2f  yards. 

30. 

91  yards. 

63. 

6  yr.  2  mo. 

10. 

.05625. 

6. 

$12.40. 

31. 

10  da.  10  hr. 

64. 

31  wk.  5  da. 

11. 

.105. 

7. 

1  mile. 

32. 

12  T.    1124 

65. 

6    mi.     177 

12. 

.288. 

8. 

$  2.58. 

Ib. 

rd. 

13. 

36.4. 

33. 

100  rods. 

14. 

.088.                   Page  327. 

Page  330. 

15. 

.605. 

1. 

17  days. 

Page  329. 

66. 

14  yd.  2  it. 

2. 

\  week. 

34. 

109    gal.    2 

67. 

145    gal.    3 

Page  323. 

3. 

$216. 

qt. 

qt. 

1. 

$562.68. 

4. 

5400  min. 

35. 

6  yd.  1  ft. 

68. 

24  pk.  2  qt. 

2. 

105.45    sq. 

5. 

5  da.  15  hr. 

36. 

5  mi.  50  rd. 

69. 

84  bu.  3  pk. 

rods. 

6. 

18  hours. 

37. 

5  wk.  1  da. 

70. 

68  quarts. 

3. 

14  rods. 

7. 

9  hr.  36  min. 

38. 

9  bu.  1  pk. 

71. 

43  qt.  1  pt. 

ANSWERS. 


72. 

10    min.    7 

Page  334. 

4. 

.78125. 

43. 

1  167.39  10. 

sec. 

1. 

10,061,280 

5. 

.265625. 

44. 

883.2429. 

73. 

17  yr.  5  mo. 

minutes. 

6. 

.187& 

45. 

2759 

74. 

24  wk.  3  da. 

2. 

$6336.12. 

7. 

.006. 

46. 

2283.9171. 

75. 

7mi.65rd. 

3. 

7609  cords. 

8. 

.03125. 

47. 

314.7032. 

76. 

25  yd.  1  ft. 

4. 

$6934.89+. 

9. 

XM625. 

48. 

1013.0418. 

77. 

64  gal.  2  qt. 

5. 

$  1833.72. 

10. 

.0035. 

49. 

639.7105. 

78. 

24  bu.  3  pk. 

6. 

$70.20. 

11. 

.24. 

50. 

49.21619. 

79. 

37  qt.  1  pt. 

7. 

$608. 

12. 

.056. 

51. 

563.7625. 

80. 

9  bu.  1  pk. 

8. 

$  101,790. 

13. 

2.875. 

4qt. 

9. 

3192;  45. 

14. 

2.9375. 

Page  339. 

81. 

2. 

10. 

$1200. 

15. 

.044. 

52. 

188.26. 

82. 

5. 

11. 

1.955  yards. 

16. 

.0016. 

53. 

288.3623. 

83. 

8. 

12. 

7960. 

17. 

5.859375. 

54. 

999.999. 

84. 

9. 

13. 

$2.35f. 

18. 

.1015625. 

55. 

13.615. 

85. 

10. 

19. 

.00390625. 

56. 

184.7569. 

Page  335. 

20. 

.013671875. 

57. 

15.0885. 

1. 

378  sq.  yd. 

14. 

$  2.56. 

21. 

.0009765625 

58. 

1999.96875. 

2. 

378  sq.  yd. 

15. 

18  pounds. 

59. 

113.1991. 

3. 

6  sq.  yd. 

16. 

327  feet. 

Page  338. 

60. 

17.84375. 

4. 

893  sq.  yd. 

17. 

$  17.88. 

22. 

dta- 

61. 

1503.5975. 

5. 

5963  sq.  yd. 

19. 

62  days. 

23. 

&- 

62. 

79.2. 

6. 

396  sq.  yd. 

20. 

$2.81. 

24. 

*Vff- 

63. 

.045264. 

7. 

288  sq.  yd. 

21. 

$2. 

25. 

H- 

64. 

1850.3125. 

8. 

36  sq.  yd. 

22. 

48     geogra- 

26. 

A. 

65. 

4.566. 

9. 

240  sq.  yd. 

phies. 

27. 

Hi 

66. 

.13875. 

10. 

36  sq.  yd. 

23. 

96. 

28. 

f 

67. 

.009438. 

29. 

t- 

68. 

6.3784. 

Page  336. 

30. 

125' 

69. 

16.93542. 

Page  331. 

24. 

$340.30. 

31. 

ToW' 

70. 

.45953125. 

13. 

420  sq.  in. 

25. 

21  bushels. 

32. 

T^V 

71. 

45.78644. 

14. 

32  sq.  yd. 

26. 

11,608  hot. 

33. 

Toffff- 

72. 

25.327+. 

15. 

256  sq.  yd. 

27. 

192  pounds. 

34. 

whfirB- 

73. 

81519.856+. 

16. 

1500  sq.  ft. 

28. 

$7*06. 

35. 

|f. 

74. 

.222+. 

17. 

270  sq.  ft. 

29. 

6. 

36. 

TffoTTr 

75. 

.321+. 

30. 

60. 

37. 

Hi 

76. 

88.4507+. 

38. 

A- 

77. 

23.328+. 

Page  332. 

Page  337. 

39. 

&• 

78. 

2626.595+. 

18. 

96  sq.  in. 

1. 

.00125. 

40. 

ToTT 

79. 

.2025+. 

19. 

72  cents. 

2. 

.025. 

41. 

135' 

80. 

.0655+. 

20. 

$40. 

3. 

.08. 

42. 

304.134. 

81. 

16.841+. 

10 


ANSWEKS. 


82. 

544.382+. 

18. 

$  125.66. 

3. 

1850  sq.  in. 

10. 

29. 

83. 

9.245+. 

19. 

$  1580.25. 

4. 

4788  sq.  in. 

11. 

$  6000. 

84. 

5.343+. 

20. 

$  35.40. 

5. 

31,104sq.in. 

12. 

6  hr.  49  min. 

85. 

.0438+. 

21. 

9.96. 

6. 

14,256  sq.  in 

45  sec.  A.M. 

86. 

60.331+. 

22. 

.0002075. 

7. 

810  sq.  in. 

13. 

A,  $  3600  ; 

87. 

304.977+. 

23. 

2.292. 

8. 

5980  sq.  in. 

B,  $2400. 

88. 

15.472+. 

24. 

26. 

9. 

6264  sq.  in. 

14. 

$  1.35. 

89. 

17.426+. 

25. 

9.2. 

10. 

8424  sq.  in. 

15. 

45  sq.  yd. 

90. 

74.3802+. 

26. 

900. 

11. 

432  sq.  ft. 

16. 

Increased  T^. 

91. 

88.537+. 

27. 

5.67. 

12. 

12  sq.  ft. 

17. 

$12. 

28. 

.01008. 

13. 

432  sq.  ft. 

18. 

360  oranges. 

Page  340. 

29. 

.33375. 

14. 

12  sq.  ft. 

19. 

$68.40. 

1. 

.034375. 

30. 

.04375. 

15. 

14  sq.  ft. 

20. 

$750. 

2. 

1200. 

31. 

* 

16. 

12  sq.  ft. 

3. 

407,294$$|£. 

32. 

i- 

17. 

14  sq.  ft. 

Page  350. 

4. 

70,234,730,841. 

33. 

k* 

18. 

437$  sq.  ft. 

1. 

$  15,373.84. 

34. 

f 

19. 

14  sq.  ft. 

2. 

$15,697.16. 

Page  341. 

35. 

TT5- 

20. 

1755  sq.  ft. 

3. 

$40,525.88. 

5. 

$444.75. 

36. 

ft. 

21. 

450  sq.  yd. 

6. 

6. 

37. 

TV- 

22. 

20  sq.  yd. 

349,129 

7. 

.2955. 

38. 

1 

23. 

108  sq.  yd. 

pupils. 

39. 

rhr- 

24. 

15  sq.  yd. 

1. 

$152.50. 

40. 

ft 

25. 

18  sq.  yd. 

Page  351. 

2. 

$  7.22. 

41. 

» 

26. 

24  sq.  yd. 

1. 

4f$  bushels. 

3. 

$  136.08. 

42. 

ft 

27. 

5^_  Sq  yd 

2. 

$13,691.16. 

4. 

$836.02$. 

28. 

3$f  sq.  yd. 

3. 

Tea,    3555f 

5. 

$  12.75. 

Page  343. 

29. 

7$  sq.  yd. 

lb.;  coffee,  6000  Ib. 

6. 

$  2392.39. 

4. 

5229  miles. 

30. 

90  sq.  yd. 

sugar,  37,012^  Ib. 

7. 

$111.45. 

5. 

$  cent. 

$  4,080  remaining. 

8. 

$  26.23$. 

6. 

$  1,303,095.17 

.  Page  348. 

4. 

68.81495  Ib. 

9. 

$  157.50. 

7. 

38  clerks. 

1. 

270  sq.  ft. 

5. 

Lost$45.97i 

10. 

$31.50. 

2. 

5  Ib.  14  oz. 

6. 

$94.51. 

11. 

$  579.72. 

Page  344. 

3. 

$  127.32. 

7. 

$  70.20. 

12. 

$47.41. 

8. 

$1914.65. 

4. 

20%cents. 

8. 

24.75  tons. 

13. 

$546.48. 

9. 

34,888  pack- 

5. 

f* 

9. 

204.0278267. 

14. 

$1129.11. 

ages. 

6. 

15. 

$  644.62. 

10. 

21,781.53696. 

7. 

M- 

Page  352. 

8. 

4581T9T  sec. 

1. 

233,675. 

Page  342. 

Page  345. 

2. 

64,725. 

16. 

$433.07. 

1. 

1512  sq.  in. 

Page  349. 

3. 

101,537$. 

17. 

$  1787.95. 

2. 

1278  sq.  in. 

9. 

216. 

4. 

216,100. 

ANSWERS. 


11 


5.    1,015,375. 

24.   301,392. 

4.   97$$. 

43.   43$. 

6.    2,336,750. 

25.   474,300. 

5.    206$f$. 

44.      5y$j. 

7.   23.367J. 

26.    385,600. 

6.    240$$. 

45.   49*. 

8.    701,025. 

27.    1,497,300. 

7.   152*. 

46.   V^ 

9.   432,200. 

28.    2,300,400. 

8.    I'll*. 

47.   |. 

10.   243,300. 

29.   324,000. 

9.   829$$. 

48.   3*. 

11.   428,400. 

30.   2,984,800. 

10.  224$$. 

12.   80,250. 

Page  361. 

13.    185,100. 

1.   $54,659,- 

Page  360. 

2-   ,h. 

14.   129,000. 

886.61. 

11.    18$. 

.046875. 

15.   230,400. 

12.   75ff. 

3.  .000000140028. 

16.   21,100. 

Page  355. 

13.   36*. 

4.  $872.87. 

17.   525,500. 

2.  $145,543,- 

14.   30$$. 

5.  6*. 

18.    145,312$. 

810.71. 

15.  49ff. 

6.   .09375  bu. 

19.   24,062$. 

5.   $69.75. 

16.  42f$. 

7.    11$  pounds. 

20.    1,828,500. 

6.    12  pages. 

17.   37f£ 

8.  $22,612.50. 

7.   $13,614.07. 

18.    68$$$. 

9.  $4.05. 

Page  354. 

8.   $. 

19.   16f$. 

10.   $296.25. 

1.    107,136. 

20.   228*. 

11.  $968.88. 

2.  604,665. 

Page  358. 

21.   17$. 

12.   $20.16. 

3.   96,145. 

2.   42  gal.  2  qt. 

22.   3$. 

13.   $7335. 

4.  494,312. 

3.   637  gal.  2  qt. 

23.   210f 

14.     aoi 

6.   473,484. 

4.    25,000  times. 

24.    52$. 

15-   *. 

6.   191,597. 

5.  9$  yards. 

25.    15$. 

16.   560.22  yards 

7.   410,896. 

6.    1501  min. 

26.   3*. 

8.    1,297,479. 

7.   66  days. 

27.  248. 

Page  362. 

9.   347,332. 

8.41  hr.  15  min. 

28.   84f. 

17.    $7.96. 

10.    1,301,234. 

9.    1440  steps. 

29.   4f 

18.   .04. 

11.    113,542. 

30.   83$. 

19.  $676. 

12.   73,350. 

Page  359. 

31.    1$. 

20.   $6. 

13.    132,790. 

10.    80  rods. 

32.   3f 

21.    Gained  $8. 

14.    110,808. 

11.    1*  min. 

33.    1384$. 

22.   21  clerks. 

15.   101,085. 

12.  4  gal.  Iqt.lpt 

.34.    19ff 

23.   1280  sheep. 

16.   852,120. 

13.   8  hr.  43  min. 

35.   8*. 

24.   4  boxes. 

17.   73,072. 

14.    2  bu.  3  pk. 

36.   2*V 

25.    H. 

18.   325,815. 

5qt. 

37.   2H- 

26.   $45. 

19.    167,892. 

15.    75  rods. 

38.   5$$. 

27.  $1033.05. 

20.   304,856. 

39.  *'. 

29.   31$cente. 

21.    169,344. 

1.    109*. 

40.    $. 

22.    212,175. 

2.   25*. 

41.   31*. 

Page  363. 

23.    710,046. 

3.    146$f 

42.  5i$. 

30.    .0002009877. 

12 


ANSWEES. 


31. 

82. 

5. 

4  weeks. 

8. 

3  quarts. 

13. 

6  hr.  27  min. 

32. 

31  years. 

6. 

34  cords. 

9. 

1  wk.  3  da. 

14. 

5  bu.  1  pk. 

33. 

21^ff. 

7. 

$357.50. 

10. 

4  T.  912  Ib. 

15. 

5  min.  13  sec. 

34. 

$108. 

16. 

2  yd.  2  ft. 

35. 

399  yr.  2  mo. 

Page  366. 

Page  368. 

17. 

1  ft.  11  in. 

17  da. 

1. 

60  Ib.  15  oz. 

1. 

42  days. 

18. 

8  T.  1234  Ib. 

36. 

219  hats. 

2. 

11  yards. 

2. 

12  days. 

19. 

2  wk.  6  da. 

37. 

7  years. 

3. 

21  da.  13  hr. 

3. 

28  days. 

20. 

4  yd.  2  ft.  3 

38. 

$999. 

4. 

28   min.    14 

4. 

56  men. 

in. 

39. 

.00012. 

sec. 

5. 

33  horses. 

21. 

4.  ..-'.  .  j 

40. 

$3. 

5. 

4T.13141b. 

6. 

18  lines. 

22. 

6. 

41. 

$110. 

6. 

123  gal.  1  qt. 

7. 

900  steps. 

23. 

8. 

42. 

63  miles. 

1  pt. 

8. 

3072  bricks. 

24. 

9. 

7. 

185  pk.  5  qt. 

9. 

11  hours. 

25. 

7. 

8. 

46  bu.  1  pk. 

10. 

77  cents. 

26. 

9. 

Page  364. 

9. 

5  weeks. 

11. 

12  days. 

27. 

8. 

1. 

777  ounces. 

10. 

990  inches. 

2. 

190  yards. 

Page  369.      Page  371. 

3. 

3520  yards. 

Page  367. 

1. 

168,932. 

28. 

12. 

4. 

89  hours. 

1. 

44  Ib.  9  oz. 

5. 

15    hr.     16 

29. 

15. 

5. 

1455  seconds. 

2. 

23  yd.  1  ft. 

min.  21T9T 

30. 

13. 

6. 

17,675  Ib. 

3. 

14    hr.     14 

sec. 

31. 

16. 

7. 

180  quarts. 

min. 

6. 

$40. 

32. 

11. 

8. 
q 

600  pints. 

4. 

26  min.    13 

7. 

6.0625. 

33. 

18. 

*}• 

10. 

632  quarts. 

5. 

4  yd.   2  ft. 

Page  370. 

1. 

'1125sq.  ft. 

11. 

62  Ib.  8  oz. 

10  in. 

8. 

$130. 

2. 

432  sq.  ft. 

12. 

62  Ib.  8  oz. 

6. 

28  gal.  2  qt. 

3. 

48  sq.  yd. 

13. 

2  ft.  4  in. 

7. 

18  bu.  3  pk. 

1. 

37  Ib.  5  oz. 

4. 

12  sq.  yd. 

14. 

2  ft.  3  in. 

8. 

4  pk.  2  qt. 

2. 

22  hr.  10 

5. 

13  sq.  yd. 

15. 

3  qt.  1  pt. 

9. 

12  weeks. 

min. 

6. 

44  sq.  yd. 

16. 

2  qt.  1  pt. 

10. 

16  T.  904  Ib. 

3. 

49  T.  835  Ib. 

7. 

7975  sq.  ft. 

17. 

1  pk.  7  qt. 

4. 

69  bu.  3  pk. 

1. 

3  Ib.  9  oz. 

5. 

14  wk.  2  d. 

Page  373. 

2. 

5  yd.  2  ft. 

6. 

21  yd.  2  ft. 

1. 

$548.80. 

Page  365. 

3. 

7  hr.  10  min. 

7. 

73  minutes. 

2. 

$37.45. 

1. 

47,789f. 

4. 

33   min.   45 

8. 

19  gal.  2  qt. 

3. 

$72. 

2. 

(a)  14.75605  ; 

sec. 

9. 

22  feet. 

4. 

$187.60. 

(6)  5999.25. 

5. 

1  ft.  4  in. 

10. 

9  yards. 

5. 

$137.10. 

3. 

598  bu.  3  pk. 

6. 

18  gal.  2  qt. 

11. 

4  Ib.  9  oz. 

6. 

11  pupils. 

4. 

16*. 

7. 

21  bu.  3  pk. 

12. 

3  gal.  2  qt. 

7. 

$480. 

ANSWERS. 


13 


Page  374. 

15.   $2.55. 

5.    497^  min. 

9.    300.04; 

8.   $333. 

16.   $56.25. 

6.    $247.50. 

6.568$. 

9.    3  words. 

17.   $49.02. 

7.    135  pounds. 

10.    1*  oranges. 

10.  60  cent*. 

18.   $3.99. 

9.   23  tons. 

11.   502$  days. 

19.   $95.02. 

10.    231'  pints. 

12.    $  5833.33  J. 

Page  375. 

20.    $96.58. 

2.   $1.33f 

21.   $23.20. 

Page  385. 

3.    1550. 

22.   $189. 

Page  381. 

13.    1360  pounds. 

4.    .000007. 

23.   $568. 

1.    121$. 

14.   2178  feet 

6.   4.975. 

24.   $225. 

2.   210$. 

15.   $52.50. 

6.    2633.0045. 

25.   $589.60. 

3.   88$. 

16.   $1.12. 

7.   |6.75. 

26.   $51.30. 

4.  331*. 

17.   $4.50. 

8.    66  cents. 

27.   $62.40. 

5.   139*. 

18.   $52$. 

8.   4  quarts. 

28.    $320. 

6.    118f 

19.   $379.50. 

10.    160  acres. 

29.   $13325. 

7.   591*. 

20.   2880  tiles. 

11.    ~>\  hours. 

30.   $52.92. 

8.   382*. 

21.   $15,000; 

12.   f 

31.   $13.50. 

9.   247ff. 

$350. 

13.  M:  As 

32.    $  2.75. 

10.   263f. 

22.    f. 

A;  16%. 

33.    $55. 

11.    5*. 

23.   $877.22. 

12.   ISrfc. 

24.   724  bushels. 

Page  376. 

13.    21$. 

25.   $1828.50. 

1.   $27.56. 

Page  379. 

14.   8  If. 

2.   $31.40. 

4.    37£sq.  yd. 

16.   12«. 

Page  386. 

3.   $18.99. 

5.   900  sq.  ft. 

16.    11H- 

26.   31  pounds. 

4.   $45.52. 

6.    2  sq.ft. 

17.  12f. 

27.   5J  miles. 

7.    3J  sq.ft. 

18.  72*. 

28.   *. 

Page  378. 

8.    8J  sq.ft. 

19.   30*. 

29.  3240  bushels. 

1.   $11.46. 

9.    13  J  sq.  ft. 

20.   40f$. 

30.   $360. 

2.   $29.10. 

10.    8100  sq.  ft. 

3.   $18.77. 

Page  387. 

4.   $11.37. 

Page  380. 

Page  383. 

1-   21*. 

5.   $21.87. 

11.    5062$  sq.ft. 

i.  mttfyd. 

2.    Idfr- 

6.   $7.47. 

12.    7  J  sq.ft. 

2.   $3000. 

3.   ^. 

7.   $3.99. 

13.    750  sq.ft. 

3.   $27.56. 

4.  6|. 

8.   $22.26. 

14.   61  1  sq.  ft. 

4.   $244*. 

5.    If 

9.  $5.08. 

15.   308}  sq.  ft. 

5.    $5.83$. 

6-    H- 

10.   $7.46. 



7.   34ff 

11.   $87.99. 

1.    38*  cents. 

Page  384. 

8.   H- 

12.   $6.30. 

2.   $10.04. 

6.   $87. 

9.    1. 

13.   $13.42. 

3.    18yd.  2ft. 

7-  ¥fo;  f 

10.   15. 

14.   $64.97. 

4.    3520  rails. 

8.   $4.42. 

11.  3.679+. 

1                                                   ANSWEES. 

12. 

.005. 

15. 

7  gal.  3  qt. 

46. 

10  yd.  1  ft. 

4. 

40  da.  19  hr. 

13. 

.004375. 

Ipt, 

lin. 

55  min. 

14. 

3.78. 

16. 

34  gal.  2  qt. 

47. 

54  gallons. 

5. 

186  gal.  3  qt. 

15. 

102.390561. 

Ipt. 

48. 

2  mi.  236  rd. 

6. 

22     hr.     30 

16. 

19,700. 

17. 

17  gal.  1  qt. 

49. 

39  gal.  3  qt. 

min.      28 

Ipt. 

1  pt. 

sec. 

Page  388. 

18. 

42  gal.  3  qt. 

50. 

2  miles. 

7. 

18  T.  862  Ib. 

17. 

.125;    8. 

19. 

15  gal.  3  qt. 

8. 

9wk.  Ida. 

18. 

90. 

20. 

88  gal.  3  qt. 

Page  391. 

9. 

53  mi.    294 

19. 

1.36. 

Ipt. 

1. 

.0015  T. 

rd. 

20. 

.26285. 

2. 

-5T5  da7- 

10. 

76  yr.  8  mo. 

Q 

45  rn.in.ut6s 

11. 

866   T.   899 

1. 

A 

Page  390. 

4. 

45  minutes. 

Ib. 

2. 

TV 

21. 

633  inches. 

12. 

140  Ib.  3  oz. 

3. 

iff 

22. 

7594  yards. 

Page  392. 

13. 

38     hr.    40 

4. 

i 

23. 

2391  quarts. 

5. 

.00625  day. 

min.  2  sec. 

5. 

if- 

24. 

2507  ounces. 

6. 

$76.87i 

14. 

180  gal.  3  qt. 

6. 

H- 

25. 

2271  inches. 

7. 

3  T.  1504  Ib. 

1  pt. 

7. 

A- 

26. 

611  quarts. 

8. 

$46.87. 

15. 

137  yd.  7  in. 

8. 

iff. 

27. 

192  feet. 

9. 

14   T.  1244 

16. 

194  mi.  183 

9. 

f. 

28. 

510  pints. 

Ib. 

rd. 

10. 

f 

29. 

102  quarts. 

10. 

tiyard. 

17. 

128  yr.  4  mo. 

11. 

A- 

30. 

34,369  Ib. 

11. 

.890625  bu. 

21  da. 

12. 

m 

31. 

54,960  sec. 

12. 

3  pk.  6  qt. 

18. 

36  wk.  5  hr. 

32. 

827  hours. 

13. 

75  cents. 

19. 

22    hr.     24 

Page  389. 

33. 

120  hours. 

14. 

$6.86. 

min.  22  sec. 

1. 

131  pints. 

34. 

165  yards. 

15. 

2  qt.  1£  pt. 

20. 

17  bu.  4  qt. 

2. 

220  pints. 

35. 

40  ounces. 

16. 

iVV 

3. 

128  pints. 

36. 

52  yd.  4  in. 

17. 

$$f. 

4. 

129  pints. 

37. 

29  Ib.  11  oz. 

18. 

1  bu.  1  pk. 

Page  394. 

5. 

279  pints. 

38. 

22  bu.  3  pk. 

Iqt, 

21. 

17  Ib.  7  oz. 

6. 

252  pints. 

Iqt. 

19. 

.885  day. 

22. 

3  bu.  2  pk. 

7. 

77  pints. 

39. 

6  da.  35  min. 

20. 

5280  feet. 

5  qt. 

8. 

85  pints. 

40. 

2  T.  972  Ib. 

23. 

7  yd.  2  ft.  7 

9. 

217  pints. 

41. 

3  mi.  12  rd. 

1. 

46  Ib.  7  oz. 

in. 

10. 

39£  pints. 

42. 

14  gal.  2  qt.  Ipt.  2. 

43  bu.  2  pk. 

24. 

10  da.  9  hr. 

11. 

39  gal. 

43. 

2  hr.  38  min. 

Iqt. 

20  min. 

12. 

19  gal.  3  qt. 

3  sec. 

25. 

3  gal.  2  qt. 

13. 

51  gal. 

44. 

27  bu.  1  pk. 

Page  393, 

Ipt. 

14. 

162    gal.   3 

5qt. 

3. 

34  yd.  2  ft. 

26. 

13     hr.    44 

qt. 

45. 

93  Ib.  7  oz. 

6  in. 

min.  30  sec. 

ANSWERS. 


15 


27. 

246  T.  1676  Ib. 

54. 

J6    hr.     16 

78. 

3  quarts. 

102. 

11  gal.  Iqt. 

28. 

11  wk.  16  hr. 

min.      15 

79. 

1  bu.  1  pk. 

Ipt. 

29. 

16  mi.  311  rd. 

sec. 

80. 

Ihr.  lOmin. 

103. 

22  bu.  2  pk. 

30. 

11  yr.  3  mo. 

55. 

138bu.2pk. 

81. 

5  Ib.  13  oz. 

2qt. 

31. 

16  Ib.  12  oz. 

2qt. 

82. 

18  bu.  3pk. 

104. 

17  yd.  1  ft. 

32. 

8  bu.  3  pk.  6 

56. 

224  gal.  3  qt. 

7qt. 

9  in. 

qt.  1  pt. 

Ipt. 

83. 

16  yd.  2  ft. 

105. 

31   mi.    108 

33. 

8yd.  1ft.  9  in. 

57. 

202  pounds. 

9  in. 

rd.  4  yd. 

34. 

12  da.  23  hr. 

58. 

5  hr.  1  min. 

84. 

11  da.  5  hr. 

106. 

25  da.  23  hr. 

45  in  in. 

57  sec. 

19  min. 

48  min. 

35. 

56  gal.  \  pt. 

59. 

7  bu.  6  qt. 

85. 

93    gal.    3$ 

36. 

67  yr.  6  mo. 

60. 

9  gal.  2  qt. 

qt. 

37. 

42  mi.  245  rd. 

Ipt. 

86. 

5  hr.  35  min. 

Page  397. 

38. 

38  T.  546  Ib. 

61. 

54  years. 

5  sec. 

108. 

13  gal.  1  pt. 

39. 

16  Ib.  12  oz. 

62. 

94  wk.  6  da. 

87. 

22T.8251b. 

110. 

17  Ib.  3  oz. 

63. 

74T.5001b. 

88. 

2wk.4da.4 

111. 

4  T.  960  Ib. 

Page  395. 

64. 

77  yards. 

hr.  48  min. 

112. 

Imi.  HOrd. 

40. 

38  wk.  3  da. 

65. 

65  mi.   160 

89. 

18  mi.   180 

113. 

3  yr.  6  mo. 

17  hr. 

rd. 

rd. 

114. 

12  bu.  3  pk. 

41. 

10  gal.  1  qt.  1 

66. 

1  da.  14  hr. 

90. 

5  yr.  9  mo. 

2*qt. 

pt. 

18  min. 

91. 

13  bu.  3  pk. 

115. 

17  yd.  4  in. 

42. 

17  hr.  24  min. 

6qt. 

116. 

5hr.  20  min. 

35  sec. 

92. 

25  gal.  2  qt. 

10  sec. 

43. 

8  yd.  1ft.  10  in. 

Page  396. 

Ipt. 

117. 

18  gal.  1  qt. 

44. 

57  bu.  1  qt. 

67. 

13J3  gal. 

93. 

33  min.  33 

Ipt. 

45. 

38  da.  18  hr. 

68. 

4  bu.  2  pk. 

sec. 

118. 

3  da.  5  hr. 

55  min. 

4qt. 

94. 

2  wk.  5  da. 

20  min. 

46. 

13  bu.  1  pk.  6 

69. 

2  hr.  34  min. 

12  hr. 

119. 

14  T.  llOlb. 

qt. 

5  sec. 

95. 

5  yd.  6  in. 

120. 

16  yd.  2  ft. 

47. 

16  gal.  2  qt.  1 

70. 

94  wk.  3  da. 

96. 

7  bu.  2  pk. 

11  in. 

pt. 

20  hr. 

Iqt. 

121. 

14  gal.  3  qt. 

48. 

6  hr.  29  min. 

71. 

20  bu.  2  pk. 

97. 

1  da.  6  hr. 

lipt. 

40  sec. 

7qt 

49  min. 

49. 

3  Ib.  10  oz. 

72. 

73  yd.  2  ft. 

98. 

3  qt.  1  pt. 

Page  399. 

50. 

25  bu.  1  pk.  7 

3  in. 

99. 

2  yd.  2  ft.  2  in 

.     1. 

1  hr.  18  min. 

qt. 

73. 

21  days. 

100. 

10  wk.  2  da. 

17  sec. 

51. 

27  bu.  3  pk.  4 

74. 

54    yr.    10 

18  hr.  15 

2. 

25  bu.  3  pk. 

qt. 

mo. 

min. 

4qt. 

52. 

76  gal.  3  qt.  1 

75. 

41  gal.  1  qt. 

101. 

50    hr.     50 

3. 

H  inches. 

pt. 

76. 

5  Ib.  3  oz. 

min.      50 

4. 

2  pk.  6  qt. 

53. 

30  Ib.  8  oz. 

77. 

7  ounces. 

sec. 

5. 

14  mi.  17  rd. 

16 


ANSWERS. 


6.    10    hr.   28 

29.    105,300. 

9.  $120. 

3.    7  o'clock  ;  3 

min. 

30.   690,300. 

10.  $912.92. 

o'clock;  5 

7.   1  ft.  10  \  in. 

o'clock. 

8.  4  hr.  43  min. 

Page  404. 

Page  408. 

4.   $18. 

30  sec. 

31.   4187fttf. 

2.    100  envel- 

9.  27  min.    10 

32.   62,132T\V7. 

opes. 

Page  412. 

sec. 

33.   9555|fff-§. 

3.   24  rugs. 

5.   25  cases. 

10.   $37.50. 

34.  9593J$H$- 

4.    72  boards. 

6.   21  posts  ;    2 

11.   9  If  cents. 

OK       Kp  (WQ  8054 

posts  ;      3 

12.   202|  miles. 

Page  409. 

posts. 

7.    1944  bricks. 

7.   31  days  ;  29 

Page  405. 

8.   72  tiles. 

days. 

Page  403. 

10.    2aV 

9.   240  boards. 

8.  43  days. 

1.   1129f 

11-    A- 

10.   264,000 

9.   23  chapters. 

2.   10,665. 

12.   14f. 

stones. 

10.  27  problems. 

3.   8077TV 

13.   7f 

11.   4816  sq.  yd. 

4.   28,813f. 

14.   71.01. 

12.   4840  sq.  yd. 

Page  413. 

5.   31,523i 

15.   .89575. 

13.   80  by    121, 

1.   238  days. 

6.   61,903^T. 

16.   148.28125. 

40  X  242, 

2.   140  days. 

7.   206,783^. 

17.   .2. 

etc. 

3.   109  days. 

8.  403,270. 

18.   $31,370.38. 

14.   16  times. 

4.    76  days. 

9.   834,085f. 

19.   1  cwt.   3  qr. 

15.    9000  sq.  ft. 

5.    151  days. 

10.   15,940,572. 

lOlb.lOoz. 

6.   284  days. 

11.   775,665. 



Page  410. 

7.   179  days. 

12.   933,273. 

1.  $216,671,399,- 

16.   41,400    sq. 

8.    139  days. 

13.   601,227. 

071.35. 

ft.;    8600 

9.   91  days. 

14.   542,817. 

sq.  ft. 

10.   151  days. 

15.   2,758,239. 

Page  406. 

17.  9400  sq.ft. 

11.   $196. 

16.   8,296,695. 

6.  $24.90. 

18.  $2800; 

12.   235  days. 

17.   1,232,766. 

7.   21,945  cu.  in. 

$330. 

13.   Tuesday. 

18.   3,855,141. 

8.  $175. 

14.   66  days. 

19.   9,733,680. 

9.  $  10  gain. 

Page  411. 

15.   August  15. 

20.   7,467,570. 

10.   $2.47. 

21.    160  sq.  yd. 

16.    170  days. 

21.   67,100. 

22.    160  rods. 

22.  310,700. 

Page  407. 

23.   64  yards. 

Page  414. 

23.   108,662^. 

4.   20  pounds. 

24.    18   sq.  yd.; 

17.   44  yr.  4  mo. 

24.   324,133^. 

5.   81. 

20£  yd. 

12  da. 

25.    113,437£. 

6.   $127,581,911,- 

25.   59TV  sq.  yd. 

18.  4  yr.  1  mo. 

26.    216  500. 

264.12. 

11  da. 

27.   426,300. 

7.   $1.03  loss. 

1.    18  hours. 

19.   8  yr.  4  mo. 

28.   2150. 

8.   4800  steps. 

2.   31  days. 

14  da. 

20.   128  yr.  2  mo. 
9  da. 
21.  Mar.  4,  1841. 
22.  33  yr.  1  mo. 
8  da. 
23.  3  yr.  9  mo. 

ANSWERS. 

8.   $1.                   18.   18,755  8q. 
9.  $108.                          yd. 
10.  $9.60. 
Page  422. 
Page  418.          19.   1  acre. 
1.   tfl                  20.    160  sq.  in. 

17 

22.  4  rd.  5  yd.  1 
ft. 
23.  6  rd.  4  yd.  1 
ft.  1  in. 
24.    14  rd.  2  ft. 
25.  5  rd.  3  yd.  1 

TS  da 

2      1  1   milpc 

ftfi  in 

1«J  Ucl. 

24.  49  yr.  3  mo. 

ft*       1  J.    Ill  11(70. 

3.  3  bu.  7  qt. 

1.  712rrods. 

.  o  in. 

26.    7rd.2yd.  1 

15  da. 

4.  A,  $750;  B, 

2.  7  rd.  41  yd. 

ft. 

26.  July  21,  '61. 

$500;   C, 

3.  7  rd.  4  yd. 

27.    17  rd.  2  yd. 

26.   117yr.5mo. 

$250. 

lift. 

1  ft,  3  in. 

27  da. 

5.  $39.38. 

4.  7  rd.  4  yd.  1 

28.  6rd.  2yd.  1 

6.  $2.15. 

ft.  6  in. 

ft.  6  in. 

Page  415. 

7.  $1407. 

5.   13  rd.   1  ft. 

29.   7  rd.  5  yd. 

1.  $34.56. 

8.  $7.05. 

6  in. 

10  in. 

2.   $15.30. 

9.   48  tiles. 

6.    12  rods. 

30.   990  inches. 

3.   469  bushels. 

10.  $75.60. 

31.  5  rods. 

4.   108  cows. 

11.  $216.66f. 

32.   1422  inches. 

6.   216  yards. 

12.   12  bu.  3  pk. 

Page  423. 

33.  7  rd.  1  yd. 

6.   192  soldiers. 

4qt. 

7.   8  rd.  5  yd. 

7.   116  gallons. 

8.  8rd.  5yd. 

8.  31  cents. 

Page  419. 

9.  8  rd.  5  yd. 

9.  $39. 

1.   18,500  sq.ft. 

10.  8rd.  5yd. 

Page  424. 

10.  $1.56. 

2.   28  sq.  yd. 

11.   9  rd.  1  ft  6 

34.    17  rd.  3  yd. 

11.  39  cents. 

in. 

35.   15  rd.  2  yd. 

12.   240  hours. 

Page  420. 

12.  9  rd.  1  yd. 

2  ft.  6  in. 

13.   2hr.  24min. 

4.   864  bricks. 

1  ft.  6  in. 

36.   13  rods. 

14.  $1.77^. 

6.   1728  bricks. 

13.   9  rd.   1  yd. 

37.  22  rd.  3  yd. 

15.  $2.20. 

6.   112  sq.  in. 

1  ft.  6  in. 

2  ft,  6  in. 

16.   27  days. 

7.   36  sq.ft. 

14.   9  rd.  1  yd. 

38.  5rd.5yd.6 

8.   45  rolls. 

1  ft.  6  in. 

in. 

Page  416. 

9.  24  lots. 

15.  9  rd.  1  yd. 

39.   12  rd.  4  yd. 

1.  $34.40. 

11.   9000  sq.ft. 

1  ft.  7  in. 

6  in. 

2.  48  cents. 

12.  $1200. 

16.  9  rd.  2  yd. 

40.  23  rd.  2  yd. 

3.  $15.60. 

1  ft.  6  in. 

6  in. 

4.  $3.90. 

Page  421. 

17.  9  rods. 

41.   Ill  rd.  lyd. 

5.  35  cents. 

13.   16  fields. 

18.  9  rods. 

6  in. 

14.   275  yards. 

19.  9  rods. 

42.   3rd.  4  yd.  2 

Page  417. 

15.   120  sq.  rd. 

20.  9  rods. 

.   ft.  6  in. 

6.  $10.40. 

16.  5  acres. 

21.   7  rd.  2  yd. 

43.  3  rd.  4  yd.  1 

7.  $22.40. 

17.  405sq.yd. 

2  ft.  1  in. 

ft.  6  in. 

18 


ANSWERS. 


Page  425. 

Page  429. 

Page  432. 

6.  $10.55. 

5.   70  cu.  in. 

2.   $46.55. 

15.  $16.50. 

320  feet. 

8.    46,656  cu.  in. 

3.  $7.32. 

16.   $25. 

7.  $9.37£. 

9.  |  cu.  yd. 

4.   795  minutes. 

17.   149fj  gal. 

8.   141. 

10.   8  ft.  X  4  ft.  ; 

5.   1  mi.  85  rd. 

18.   $2.51. 

10.  $4.50. 

16ft.x2ft.; 

2  ft.  6  in. 

19.   2500. 

ST¥T  tonp- 

etc. 

7.   40  sq.  in. 

21.   $6.66. 

11.   T8T  acre. 

11.   5184  cu.  ft. 

8.   .625  year. 

24.   23^. 

12.   3x7x11; 

643  Ib.  9.6 

25.  $2.80. 

Page  438. 

6x7x5£; 

oz. 

12.    $6;     $18; 

etc. 

10.   31.416. 

Page  433. 

$24;  $36 

13.   1  cu.  ft. 

2  inches. 

i.  **;!**• 

13.   672  hens. 

smaller. 

2.   .571f  ;  .625. 

14.  51f$  miles. 

14.  $132. 

1.  *3&. 

3.   199.925. 

15.  $114.60. 

4.   .012. 

16.   $1250. 

Page  430. 

5.  $27. 

17.   106,294.4. 

Page  426. 

3.   $28,800. 

6.   36  spoons. 

3757.2. 

15.   3  feet. 

5.   f  ton. 

18.  54  sq.  yd. 

16.   About  *l\  gal. 

7.  $75.47. 

Page  434. 

160  sq.  in. 

17.   About  l£cu.  ft. 

9.  40  bushels. 

7.  $20. 

19.   $400. 

18.   30  gallons. 

10.  |. 

8.  $71.28. 

20.   1232  mi.  ; 

19.  30  bushels. 

9.  4  feet. 

1730  yd.  ; 

20.   9  cords. 

1.   $693. 

10.  $120. 

1,020,304 

21.    162,000  bricks. 

2.   45  cents. 

11.    76  sq.  yd. 

Ib. 

10,368,000  cu. 

3.   iff  day. 

12.   $24.75. 

21.  $24.374. 

in. 

4.   $2.39. 

13.  $23.52. 

22.   173,218.35; 

22.   31,104  bricks. 

5.    24  miles. 

14.   12,960  Ib. 

814.43. 

23.   27  bricks. 

15.   $14.28. 

24.   40,000  bricks. 

Page  431. 

Page  439. 

25.  $2048. 

1.  $136.57. 

Page  435. 

23.  3300ft.; 

2.   60ff  acres. 

3.    lOtf;  84f. 

flday; 

3-   dfifr- 

4.  41.00679. 

110,672 

Page  428. 

4-  5|f. 

5.   2750  sq.  yd. 

oz. 

4.    1562.5. 

5.   Increased  j1^. 

6.   926J. 

24.   31f£sq.yd.; 

5.   $39.81. 

6.   8.384964. 

42  feet. 

8.    88  cents  ;  ^fa 

9.   47  min.  12& 

Page  437. 

25.  f. 

rod. 

sec. 

1.  $447.77f. 

26.  $714. 

9.   &bbl.:    127$ 

10.  H*. 

2.  $493.76^. 

27.  $9120; 

cu.  yd. 

11.  $7.41|. 

3.  $24.75. 

$  14,820. 



13.  453$  miles. 

4.   $141.95. 

28.  yfo;  278  A- 

1.  $3.62. 

14.   99||  cents. 

5.   100. 

29.  $166.72. 

ANSWERS. 


19 


30.  $60.75. 

38.   22ff 

28.  $518.40. 

14.    30;  15;  135. 



39.  30^. 

29.  $183. 

15.  9  pounds. 

1.  38f}. 

40.  20^. 

16.   19  rods. 

2.  S7&. 

Page  445. 

17.   85  feet. 

3.  58}}- 

Page  441. 

1.   150  sq.  in. 

4.  31H- 

1.  460.12. 

2.  1536  sq.  yd. 

Page  450. 

5.  61&. 

21,355.74. 

3.    117  sq.  yd. 

18.   Son,  $  40  ; 

6.   bo-ff. 

2.   .4551. 

4.    1225  sq.ft. 

daughter, 

7.  66^. 

3.  ^fc. 

5.   1554  sq.  yd. 

$80. 

8.  95H- 

4-  HI 

6.   6111  sq.  ft. 

19.   25  days. 

9.  38}f. 

5.   11  2%. 

7.   924  sq.  m. 

20.  Girl,  $80; 

10.   89^. 

6.   725}f}. 

8.   81  sq.  ft. 

boy,  $40. 

7.  -jJ^tj.. 

21.   Father,  30  da.; 

Page  440. 

8.   .675. 

Page  446. 

son,  15  da. 

11.  6|. 

9.  $227.60^. 

9.   81  sq.  ft. 

22.  3    dimes,    6 

12.   13H- 

10.  $3800. 

10.   735  sq.  yd. 

nickels,     18 

13.  54}$. 

11.   23hr.2min. 

cents. 

14.  67f|. 

8}  sec. 

Page  448. 

23.   25  yards. 

15.   17}$. 

12.  $11.08}. 

2.  20  and  80. 

24.   25   rods;    100 

16.  61$. 

13.   1$  yards. 

3.  $2000; 

rods. 

17.  37H- 

14.   1. 

$4000; 

25.  Speller,  15  f.  ; 

18.   18ff. 

15.  $187.50. 

$  12,000. 

reader,  45  f. 

19.  39}f. 

16.   7  days. 

4.   18  girls;  36 

26.  60  and  12. 

20.  25*}. 

boys. 

27.   18  nuts  ;  9 

21.   131$. 

5.   13  and  65. 

nuts  ;  27  nuts. 

22.   211$. 

Page  442. 

6.    13. 

23.  663|. 

17.   216  sq.  in.; 

Page  452. 

24.   185/j. 

1}  sq.  ft.  ; 

Page  449. 

1.   24. 

25.   I03}f. 

216  cu.  in.  ; 

7.   11. 

2.   24. 

26.  95J. 

}  cu.  ft. 

8.  $3000; 

3.  42. 

27.  81$. 

18.   28}  feet. 

$6000; 

4.   84. 

28.  98|. 

19.   10|$  years. 

$  18,000. 

5.  24. 

29.  513}f 

20.   $21.67}. 

9.   12  and  60. 

6.   70. 

30.  431&. 

21.  $199.50. 

10.  9  marbles  ; 

7.   72. 

31.    15J}. 

22.  7.92  inches. 

18  marbles  ; 

8.  40. 

32.   13$f. 

23.  40  cents. 

27  marbles. 

9.  360. 

33.   16f}. 

24.  $1800; 

11.   36  years; 

10.   160. 

34.   12|}. 

$6300. 

6  years. 

11.   18. 

35.  21^. 

25.  $874.80. 

12.  8. 

12.   18. 

36.  31^V. 

26.  4142$  Ib. 

13.    1;    4;     12; 

13.  8. 

37.  23^. 

27.  62}  % 

24. 

14.  16. 

20 


ANSWERS. 


15.    12. 

Page  454. 

Page  456. 

8.    62  years. 

16.    20. 

10.   60;  420. 

9.   27. 

9.  84;  12. 

17.   900. 

11.   540;  18. 

10.   3. 

10.   $108. 

18.   60. 

12.   9. 

11.   28. 

11.    17;  28. 

19.   60. 

13.   20  peaches  ;  5 

12.   96. 

12.    $16;   $11. 

20.   32. 

plums. 

13.   144. 

13.  Cows,  $45; 

14.   $200;  $600; 

14.    18. 

horses, 

$  700. 

15.   24. 

$125. 

Page  453. 

15.  $60;  $140. 

16.   6. 

14.   3    dimes;    14 

21.   36. 

16.  $300. 

17.   32. 

half  dimes. 

22.   222. 

17.   64  marbles. 

18.   18. 

15.   74  and  26. 

23.   180. 

18.    $2;  $3;  $10. 

19.   12. 

16.   21   boys;    33 

24.   72. 

19.    $4;   $2. 

20.   20. 

girls. 

25.  320. 

20     3    horqps-    12 

26.   7. 

/jw.      O     llUIotJo  ,      Lpi 

cows. 

2.    15. 

3.   9. 

Page  458. 

1.   15  and  75. 

Page  455. 

4.  15  marbles;  33 

17.    $3600;  $6000; 

2.   28f;  71f. 

1.   19. 

marbles. 

$  8400. 

3.  $816. 

2.   22. 

18.    44;    11. 

4.  $180. 

3.  47. 

Page  457. 

19.   5  five-cent 

5.   89. 

4.   14. 

5.   25  ft.;  100  ft. 

stamps  ;     20     two- 

6.    100. 

5.   9. 

6.   39  acres  ;  47 

cent    stamps  ;      35 

7.  40;  15. 

6.   10. 

acres. 

postal  cards. 

8.  f|. 

7.  6. 

7.    1059  votes; 

20.   8  horses;    25 

9-   If- 

8.  33. 

1377  votes. 

cows  ;  55  sheep. 

HISTORY. 


Sheldon's  General  History.  For  high  school  and  college.  The  only  history  fa 
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Sheldon's  Greek  and  Roman  History.  Contains  the  first  250  pages  of  the  above 
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Poacher's  Manual  to  Sheldon's  History.  Puts  into  the  instructo.  's  hand  the  key 
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Bridgman's  Ten  Years  of  Massachusetts.  Pictures  the  development  of  the 
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Shumway's  A  Day  in  Ancient  Borne.  With  59  illustrations.  Should  find  a  place 
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Old  South  Leaflets  on  TJ.  S.  History.  Reproductions  of  important  political  and 
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This  general  series  of  Old  South  Leaflets  now  includes  the  following  subjects  : 
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Lincoln's  Inaugurals  and  Emancipation  Proclamation,  The  Federalist,  Nos.  i 
and  2,  The  Ordinance  of  1787,  The  Constitution  of  Ohio,  Washington's  Letter  to 
Benjamin  Harrison,  Washington's  Circular  Letter  to  the  Governors.  (38  Leaflets 
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Allen's  History  Topics.  Covers  Ancient,  Modern,  and  American  history,  ano1 
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Fisher's  Select  Bibliog.  of  Ecclesiastical  History.  An  annotated  list  of  the 
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Hall's  Methods  of  Teaching  History.  "  Its  excellence  and  helpfulness  ought  to 
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Wilson's  The  State.  Elements  of  Historical  and  Practical  Politics.  A  text-booV 
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Sanford's  Laboratory  Course  in  Physiological  Psychology.     The  course 

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Lange's  Apperception :  A  monograph  on  Psychology  and  Pedagogy.    Trans- 

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READING. 


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Smith's  Reading  and  Speaking.  Familiar  Talks  to  those  who  would  speak  well  in 
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NUMBER. 


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At WOOd'S  Complete  Graded  Arithmetic.  Present  a  carefully  graded  course  in 
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Sutton  and  Kimbrough's  Pupils'  Series  of  Arithmethics. 

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Safford's  Mathematical  Teaching.  Presents  the  best  methods  of  teaching,  from 
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Badlam's  Aids  tO  Number.  For  Teachers.  First  Series.  Consists  of  25  cards  for 
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ALGEBRA  AND  GEOMETRY. 

Academic  Algebra. 

By  E.  A.  BOWSER,  Prof,  of  Mathematics,  Rutgers  College.     Half  leather. 
366  pages.     Price  by  mail,  $1.25. 

This  work  is  designed  as  a  text-book  for  common  and  high  schools 
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through  the  Progressions,  and  including  Permutations  and  Combinations 
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College  Algebra. 

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Plane  and  Solid  Geometry. 

Half  leather.    402  pages.     Price  by  mail,  $1.40.     Introduction  price,  $1.25. 

This  work  combines  the  excellences  of  Euclid  with  those  of  the  best 
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(2)  to  discipline  and  invigorate  the  mind  —  to  train  it  to  habits  of  clear 
and  consecutive  reasoning. 

Hopkins     Plane   Geometry,  on  the  Heuristic  Plan. 

By  G.  I.  HOPKINS,  High  School,  Manchester,  N.H.    Boards.    60  cents. 
The  demonstrations  are  purposely  incomplete,  so  that  the  pupil  is 
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ARITHMETIC. 

Aids  to  Dumber.  —  First  Series.     Teach^  Edition. 


Oral  Work  —  One  to  ten.     25  card*  with   concise   directions.      By  ANNA   B. 
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to  Dumber.  —  First  Series.    Putts'  Edition. 

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Problems.      By  Miss  H.  A.  LUDDINGTON, 


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{Mathematical  Teaching  and  its  {Modern  {Methods. 

By  TRUMAN   HKNRY  SAFFC 
Mass.    Paper.     47  pages.    1 

Tbe  New  Arithmetic. 


By  TRUMAN   HKNRY   SAFFORU,  Ph.  D.,   Professor  of  Astronomy,   Williams   College, 
Mass.    Paper.     47  pages.    Retail  price,  25  cents. 


By  300  authors.     Edited  by  SKYMOUR  EATON,  with  Preface  by  T.  H.  SAFFORD,  Pro- 
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MA  THEM  A  TICS. 


Bowser's  Academic  Algebra.  A  complete  treatise  through  the  progressions,  inelod 
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Bowser's  Plane  and  Solid  Geometry.  Combines  the  excellences  of  Euclid  with 
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Bowser's  Plane  Geometry.    Half  leather.   85  cts. 

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Hanus's  Geometry  in  the  Grammar  Schools.    An  essay,  together  with  illustrative 

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Hopkin's  Plane  Geometry.    On  the  heuristic  plan.     Half  leather.    85  cts. 

Hunt's  Concrete  Geometry  for  Grammar  Schools.  The  definitions  and  ele- 
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Waldo's  Descriptive  Geometry.  A  large  number  of  problems  systematically  ar 
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The  New  Arithmetic.  By  300  teachers.  Little  theory  and  much  practice.  Also  or 
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DRAWING  AND  MANUAL  TRAINING. 


Johnson's  Progressive  Lessons  in  Needlework.     Explains  needlework  from  its 

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Seidel's  Industrial  Instruction  (Smith).  A  refutation  of  all  objections  raised  ?«ainst 
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Thompson's  Educational  and  Industrial  Drawing. 

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Mechanical  Manual.     172  pages.     Paper.     75  cts. 
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Solids.  Illustrated.  66  pages.  Large  8vo.  Paper.  30  cts. 

Thompson's  Manual  Training,  NO.  2.  Treats  of  Mechanical  Drawing,  Clay- 
Modelling  in  Relief,  Color,  Wood  Carving,  Paper  Cutting  and  Pasting.  Illustrated. 
70  pp.  Large  8vo.  Paper.  30  cts. 

Waldo's  Descriptive  Geometry.  A  large  number  of  problems  systematically  ar- 
ranged, with  suggestions.  85  pages.  90  cts. 

Whitaker's  HOW  to  Use  WOOd  Working  Tools.  Lessons  in  the  uses  of  the 
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Woodward's  Educational  Value  of  Manual  Training.    Sets  forth  more  clearly 

and  fully  than  has  ever  been  done  before  the  true  character  and  functions  of  ma^ul  train 
ing  in  education.    96  pages.     Paper.    25  cts. 

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Why  should  Teachers 


"°  man  caw  stand  high  in  any   profession  who   is  not  familiar 
aitn  its  nist0ry  and  literature. 

'*  saves  time  which  might  be  wasted  in  trying  experiments  that 
t,aue  aireacfy  oeen  tried  and  found  useless. 

Compayre'S  History  Of  Pedagogy.  "  The  best  and  most  comprehensive 

history  of  Education  in  English."  —  Dr.  G.  S.  HALL  ......  $'-75 

Compayre'S  Lectures  On  Teaching.  "  The  best  book  !  n  existence  on 

the  theory  and  practice  of  Education."  —  Supt.  MACALLISTER,  Philadelphia.  .  1.75 

Gill's  System  Of  Education.  "It  treats  ably  of  the  Lancaster  and  Bell 

movement  in  Education  —  a  very  important  phase."  —  Dr.  W.  T.  HARRIS.  .  1.25 

RadestOCk'S  Habit  in  Education.  "  It  will  prove  a  rare  '  find  '  to  teach- 
ers who  are  seeking  to  ground  themselves  in  the  philosophy  of  their  art."  — 
E.  H.  RUSSELL,  Worcester  Normal.  .  ........  0.75 

Rousseau's  Emile.  "  Perhaps  the  most  influential  book  ever  written  on  the 

subject  of  Education."  —  R.  H.  QUICK.  ........  0.90 

Pestalozzi's  Leonard  and  Gertrude.  "  If  we  except  '  Emile  '  only,  no 
more  important  educational  book  has  appeared,  for  a  century  and  a  half,  than 
'  Leonard  and  Gertrude.'  "  —  The  Nation.  .......  0.90 

Richter's  Levana;  or  the  Doctrine  of  Education.    "A  spirited 

and  scholarly  book."  —  Prof.  W.  H.  PAYNE.      ...        .        .        .        .         1.40 

Rosmini'S   Method    in    Education.     "The  most  important  pedagogical 

work  ever  written."  —  THOMAS  DAVIDSON.         .......         1.50 

Malleson's  Early  Training  of  Children.  "  The  best  book  for  mothers 

I  ever  read."  —  ELIZABETH  P.  PEABODY.          .......        o-75 

Hall's  Bibliography  of  Pedagogical  Literature.    Covers  every 

department  of  Education.      .         .        .        .        .        .        .        .        .        *        .        i-5° 

Peabody's  Home,  Kindergarten  and  Primary  School  Educa- 
tion. "The  best  book  outside  of  the  Bible  I  ever  read."  —  A  LEADING 
TEACHER  ...............  i.oo 

Newsholme'S  School  Hygiene.     Already  in  use  in  the  leading  training 

colleges  in  England.       .        .  .       •»        .        .        .        .        .        •        .        °-7S 

DeGarmo's  Essentials  of  Method.  "  It  has  as  much  sound  thought  to 
the  square  inch  as  anything  I  know  of  in  pedagogics."  —  Supt.  BALLIET, 
Springfield,  Mass.  ............  0.65 

Hall's   Methods  Of  Teaching  History.     "  Its  excellence  and  helpful- 

ness ought  to  secure  it  many  readers."  —  The  Nation  ......         1.50 

Seidel'S  Industrial  Education.  "It  answers  triumphantly  all  objections 
to  the  introduction  of  manual  training  to  the  public  schools."  —  CHARLES  H. 
HAM,  Chicago  ..............  0.90 

Badlam's  Suggestive  Lessons  on  Language  and  Reading. 
"The  book  is  all  that  it  claims  to  be  and  more.  It  abounds  in  material  that 
will  be  of  service  to  the  progressive  teacher."  —  Supt.  DUTTON,  New  Haven.  1.50 

Redway's  Teachers'  Manual  of  Geography.    "  Its  hints  to  teachers 
are  invaluable,  while  its  chapters  on  '  Modern  Facts  and  Ancient  Fancies  '  will 
be«a  revelation  to  many."  —  ALEX.    E.    FRYE,  Author  of  "The  Child  I'M 
Nature."        ..............        0.65 

Topics    in    Geography.      "  Contains  excellent  hints  and  sug- 
gestions of  incalculable  aid  to  school  teachers."  —  Oakland  (Cal.)  Tribune.      .        0.65 


D.  C.  HEATH  &  CO.,  Publishers, 

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BUSINESS. 


Heavy's  Practical  Business  Bookkeeping.     All  needless  discussion  b  carefuli, 

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Seavy's  Manual  Of  Business  Transactions.     Contains  transactions  for  practice, 
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Shaw's  Practice  Book  of  Business  Forms  and  Elementary  Bookkeeping! 

Treats  of  the  best  methods  of  keeping  simple  accounts  and  furnishes  a  necessary  knowl 
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Boards .• 24 

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Book  of  Blank  Notes,  Bill  Heads,  Bank  Checks,  Receipts,  Orders,  etc.         .        .        .20 

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The  Volpenna  Vertical  Writing  Books.    (Newiands  and  Row).   /*/rr«. 

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GEOGRAPHY  AND  MAPS. 


Heath's  Practical  School  Maps.  Each  30  x  40  inches.  Printed  from  new  pfatet 
and  showing  latest  political  changes.  The  common  school  set  consists  of  Hemisphere*, 
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on  wall,  singly,  $1.25  ;  per  set  of  seven,  $7.00.  Mounted  on  cloth  and  rollers.  Singly. 
$2.00.  Mounted  on  cloth  per  set  of  seven,  $12.00.  Sunday  School  set.  Canaan  and 
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Heath's  Outline  Map  Of  the  United  States.  Invaluable  for  marking  territorial 
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Historical  Outline  Map  Of  Europe.  12  x  18  inches,  on  bond  paper,  in  black  outline. 
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Jackson's  Astronomical  Geography.  Simple  enough  for  grammar  schools.  Used 
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Map  Of  Ancient  History.  Outline  for  recording  historical  growth  and  statists  (14* 
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Nichols'  Topics  in  Geography.  A  guide  for  pupils*  use  from  the  primary  through 
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Picturesque  Geography.  12  lithograph  plates,  15  x  20  inches,  apd  pamphlet  describing 
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Progressive  Outline  Maps:  United  States,  *World  on  Mercator's  Projection  (iz  x 
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Asia,  Australia,  *British  Isles,  *England,  *Greece,  *Italy,  New  England,  Middle  Atlan- 
tic States,  Southern  States,  Southern  States — western  section,  Central  Eastern  States, 
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Those  marked  with  Star  (*)  are  also  printed  in  black  outline  for  use  in  teaching  history. 

Red  Way's  Manual  Of  Geography.  I.  Hints  to  Teachers;  II.  Modern  Facts  and 
Ancient  Fancies.  65  cts. 

Redway's  Reproduction  of  Geographical  Forms.    I.  Sand  and  Clay-Modelling; 

II.  Map  Drawing  and  Projection.     Paper.     30  cts. 

Roney's  Student's  Outline  Map  of  England.  For  use  in  English  History  and 
Literature,  to  be  filled  in  by  pupils.  5  cts. 

Trotter's  Lessons  in  the  New  Geography.  Treats  geography  from  the  Kumar 
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ELEMENTARY  SCIENCE. 


Bailey'3  Grammar  SchOOl  PhysiCS.  A  series  of  inductive  lessons  in  the  elements 
of  the  science.  In  press. 

Ballard's  The  World  Of  Matter.  A  guide  to  the  study  of  chemistry  and  mineralogy; 
adapted  to  the  general  reader,  for  use  as  a  text-book  or  as  a  guide  to  the  teacher  in  giving 
object-lessons.  264  pages.  Illustrated.  Ji.oo. 

Clark's  Pr&Ctical  Methods  in  MicrOSCOpy.  Gives  in  detail  descriptions  of  methods 
that  will  lead  the  careful  worker  to  successful  results.  233  pages.  Illustrated,  f  1.60. 

Clarke's  Astronomical  Lantern.  Intended  to  familiarize  students  with  the  constella- 
tions by  comparing  them  with  fac-similes  on  the  lantern  face.  With  seventeen  slides, 
giving  twenty-two  constellations.  $4.50. 

Clarke's  HOW  tO  find  the  Stars.  Accompanies  the  above  and  helps  to  an  acquaintance 
with  the  constellations.  47  pages.  Paper.  15  cts. 

Guides  for  Science  Teaching.  Teachers'  aids  in  the  instruction  of  Natural  History 
classes  in  the  lower  grades. 


I.     Hyatt's  About  Pebbles.     26  pages.     Paper.     10  cts. 
'  i  A  Few  Common  Pla 
iommercial  and  otl 
First  Lessons  in 
25  cts. 


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III.  Hyatt's  Commercial  and  other  Sponges.    Illustrated.    43  pages.  Paper.   20  cts. 

IV.  Agassiz's  First  Lessons  in  Natural  History.     Illustrated.     64  ; 


pages.     Paper. 


V.  Hyatt's  Corals  and  Echinoderms.     Illustrated.     32  pages.    Paper.     30  cts. 

VI.  Hyatt's  Mollusca.     Illustrated.    65  pages.     Paper.     30  cts. 

Vll  H'-att's  Worms  and  Crustacea.     Illustrated.     68  pages.     Paper.     30  cts. 

VIII  H  /att's  Insecta.     Illustrated.    324  pages.    Cloth.    $1.25. 

XII  (  rosby's  Common  Minerals  and  Rocks.     Illustrated.    200  pages.     Paper,  43 

cts.     Cloth,  60  cts. 

XI  11  Richard's  First  Lessons  in  Minerals.     50  pages.     Paper.     10  cts. 

XIV  Bowditch's  Physiology.    58  pages.     Paper,    sorts. 

XV  Clapp's  36  Observation  Lessons  in  Minerals.    80  pages.     Paper.    30  cts. 

XVI  Phenix's  Lessons  in  Chemistry.     In  press. 

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